T1 and T2 are stop times, both of these events belong to Ft, their union also belongs to Ft. Hence, we can conclude that T1 + T2 is also a stop time.
We are given two stop times T1 and T2 with respect to the same filtration.
We are to prove that the maximum and the sum of T1 and T2, i.e., max(T1, T2) and T1 + T2 are also stop times.
Let us consider the stop time T1.
This means that the event {T1 ≤ t} belongs to the sigma-algebra Ft, for all t≥0.
Similarly, let us consider the stop time T2.
This means that the event {T2 ≤ t} belongs to the sigma-algebra Ft, for all t≥0.
We are to prove that max(T1, T2) is also a stop time.
We can do so by considering the following event:{max(T1, T2) ≤ t}.
If T1 ≤ T2, then this event reduces to {T2 ≤ t} which belongs to Ft.
Similarly, if T2 ≤ T1, then this event reduces to {T1 ≤ t} which also belongs to Ft.
Thus, we can conclude that max(T1, T2) is a stop time.
We are to prove that T1 + T2 is also a stop time.
We can do so by considering the following event:{T1 + T2 ≤ t}.
This event can be expressed as:{T1 ≤ t − T2} ∪ {T2 ≤ t − T1}.
To know more about event visit:
https://brainly.com/question/30169088
#SPJ11
do not send links or files!! i really need help lol
Answer:
x²+11x+30
Step-by-step explanation:
for this question, all we need to do is multiply (x+5) by (x+6)
we can use the distributive property.
(x+5)(x+6)
x² + 5x + 6x + 30
x² + 11x + 30
: Let S = {1,2,3,...,18,19). Let R be the relation on S defined by xRy means "xy is a square of an integer". For example 1R4 since (1)(4) = 4 = 22. a. Show that R is an equivalence relation (i.e. reflexive, symmetric, and transitive). b. Find the equivalence class of 1, denoted 7. c. List all equivalence classes with more than one element.
a. The relation R defined on the set S = {1, 2, 3, ..., 18, 19} is an equivalence relation. It is reflexive, symmetric, and transitive, b. The equivalence class of 1, denoted [1], consists of the perfect squares in S: {1, 4, 9, 16}, c. The equivalence classes with more than one element are [1], [2], [3], ..., [18], and [19]. Each equivalence class represents a set of numbers that are squares of integers.
a. To show that the relation R is an equivalence relation, we need to demonstrate that it is reflexive, symmetric, and transitive.
i. Reflexive: For R to be reflexive, every element in S must be related to itself. Since the square of any integer is still an integer, xRx holds for all x in S, satisfying reflexivity.
ii. Symmetric: For R to be symmetric, if xRy holds, then yRx must also hold. Since multiplication is commutative, if xy is a square of an integer, then yx is also a square of an integer. Hence, R is symmetric.
iii. Transitive: For R to be transitive, if xRy and yRz hold, then xRz must also hold. Since the product of two squares of integers is itself a square of an integer, xz is also a square of an integer. Thus, R is transitive.
b. To find the equivalence class of 1, denoted [1], we determine all elements in S that are related to 1 under R. In this case, [1] consists of the perfect squares in S: {1, 4, 9, 16}.
c. The equivalence classes with more than one element are [1], [2], [3], ..., [18], and [19]. Each equivalence class represents a set of numbers that are squares of integers. The equivalence class [1] includes all perfect squares in S, while the other equivalence classes consist of a single element, which are non-square integers.
To know more about equivalence relation, click here: brainly.com/question/30956755
#SPJ11
Can someone answer all of them? Tysm!
In a river bank
you can put the answers from the end
1.) 1/12
2.) 2/13
3.) 8/41
4.) 1/4
5.) 11/74
6.) 2/35
7.) 15/58
8.) 5/18
9.) 1/7
10.) 1/13
Consider the triple integral defined below: = f(x, y, z) dv 2y² 9 Find the correct order of integration and associated limits if R is the region defined by 0 ≤ ≤1-20≤x≤2- and 0 ≤ y. Remember that it is always a good idea to sketch the region of integration. You may find it helpful to sketch the slices of R in the zy-, zz- and yz-planes first. Hint: There are multiple correct ways to write dV for this integral. If you are stuck, try dV=dz dzdy s s s f(x, y, z) ddd I=
The correct order of integration and associated limits for the given triple integral I = ∫∫∫[f(x, y, z)] dx dy dz with limits: 0 ≤ x ≤ 1 - 2y, 0 ≤ y, 0 ≤ z ≤ 2y²
The correct order of integration and associated limits for the given triple integral, let's first examine the region of integration R and its slices in different planes.
Region R is defined by 0 ≤ z ≤ 2y² and 0 ≤ x ≤ 1 - 2y.
1.Slices in the zy-plane: In the zy-plane, z is restricted to 0 ≤ z ≤ 2y², and y is unrestricted. Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dz dy dx
2.Slices in the zx-plane: In the zx-plane, z is unrestricted, and x is restricted to 0 ≤ x ≤ 1 - 2y. Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dx dz dy
3.Slices in the yz-plane: In the yz-plane, y is unrestricted, and z is restricted to 0 ≤ z ≤ 2y². Therefore, the integral can be written as:
I = ∫∫∫ f(x, y, z) dV = ∫∫∫ f(x, y, z) dy dz dx
Considering the given hint, we can choose any of the above orders of integration as all of them are correct ways to write the integral. However, for simplicity, let's choose the order: I = ∫∫∫ f(x, y, z) dz dy dx.
Now, let's determine the limits of integration for each variable in this order:
∫∫∫ f(x, y, z) dz dy dx = ∫∫ [∫[f(x, y, z) dz] from z=0 to z=2y²] dy dx
The innermost integral with respect to z is evaluated from 0 to 2y². The next integral with respect to y is evaluated from 0 to a certain limit determined by the region R. Finally, the outermost integral with respect to x is evaluated from 0 to 1 - 2y.
Therefore, the order of integration and the associated limits for the triple integral are:
I = ∫∫∫ f(x, y, z) dz dy dx
I = ∫∫ [∫[f(x, y, z) dz] from z=0 to z=2y²] dy dx
I = ∫∫∫[f(x, y, z)] dx dy dz with limits: 0 ≤ x ≤ 1 - 2y, 0 ≤ y, 0 ≤ z ≤ 2y²
To know more about order of integration click here :
https://brainly.com/question/30286960
#SPJ4
Can somebody help me!!??
Answer:
D) 8
Step-by-step explanation:
We're looking for [tex]x[/tex], which means the 7x and 5x will have the SAME x.
7(8) = 56
For the C corner, it is cornered as a 90 degree angle, so that means the (7x) + 34 degrees NEEDS to equal 90.
7(8) = 56 + 34 = 90!!
Now we have to see if 5(8) is correct
5(8) = 40 + 50 [The degree mark] = WHICH EQUALS 90 TOO..
Therefore, the answer is D) 8
PARK is a parallelogram. Find the value of x.
Answer: 40
Step-by-step explanation:
CAN SOMEONE HELP PLS :D will mark brainliest ;)
Answer: the second one/ \/36/6
Step-by-step explanation:
What is the probability of getting a number greater than or equal to 5 when rolling a number cube numbered 1 to 6?
Answer:
There is a 1/3 (or 0.33%) probability of rolling a number greater than or equal to 5.
Step-by-step explanation:
First, find what numbers are greater than or equal to 5:
5 and 6
Find what options you can get on a number cube:
1, 2, 3, 4, 5, and 6
Out of the 6 possible outcomes, there are only 2 that will get a number greater than or equal to 5. Write this as a fraction:
2/6
Simplify:
1/3
solve the equation 3a - 6 = -12.
Answer:
a=-2
Step-by-step explanation:
3a−6=−12
Step 1: Add 6 to both sides.
3a−6+6=−12+6
3a=−6
Step 2: Divide both sides by 3.
3a/ 3 = −6/ 3
a=−2
answer the question true or false. the null distribution is the distribution of the test statistic assuming the null hypothesis is true; it is mound shaped and symmetric about the null mean .
False, the null distribution is the distribution of the test statistic assuming the null hypothesis is true; it is mound-shaped and symmetric about the null mean.
The null distribution is the distribution of the test statistic under the assumption that the null hypothesis is true. However, its shape and symmetry are not necessarily predetermined.
The null distribution can take various forms depending on the specific test and the underlying data. It may or may not be mound shaped or symmetric about the null mean. The shape and characteristics of the null distribution are determined by the specific hypothesis being tested, the sample size, and other factors.
Learn more about null distribution at
https://brainly.com/question/31692159
#SPJ4
Consider the following second order linear ODE y" - 5y + 6y = 0, where y' and y" are first and second order derivatives with respect to x. (a) Write this as a system of two first order ODEs and then write this system in matrix form. (b) Find the eigenvalues and eigenvectors of the system. (e) Write down the general solution to the second order ODE. (a) Using your result from part 3 (or otherwise) find the solution to the following equation. y' - 5y + y = 32
a. System in the matrix form is x' = Ax where A = [tex]\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex] and x = [y, u].
b. The eigenvalues of the system are λ₁ = 5 and λ₂ = 1 and eigenvector are v₁ and v₂ = v₁, and v₁ is any non-zero value.
c. The general solution is equal to y(x) = c₁ × [tex]e^{(5x)[/tex] × [v₁] + c₂× [tex]e^{(x)[/tex]× [v₂].
a. Solution to the equation. y' - 5y + y = 32 is y(x) = c₁ × [tex]e^{(5x)[/tex] + c₂ × [tex]e^{(x)[/tex].
(a) To write the second order linear ODE as a system of two first order ODEs,
Introduce a new variable u = y'.
Then, we have,
u' = y'' - 5y + 6y
= -5y + 6u
Now, write this as a system of two first order ODEs,
y' = u
u' = -5y + 6u
To express this system in matrix form,
Define the vector x = [y, u] and the matrix A = [tex]\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex]
The system can then be written as,
x' = Ax
(b) To find the eigenvalues and eigenvectors of matrix A, solve the characteristic equation,
|A - λI| = 0
where I is the identity matrix.
Substituting the values of A, we have,
[tex]|\left[\begin{array}{ccc}0&1\\-5&6\end{array}\right][/tex] [tex]-\lambda\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]|[/tex] = 0
[tex]\left[\begin{array}{ccc}-\lambda&1\\-5&6-\lambda\end{array}\right][/tex] = 0
(-λ)(6-λ) - (-5)(1) = 0
λ²- 6λ + 5 = 0
Factoring the quadratic equation, we get,
(λ - 5)(λ - 1) = 0
So the eigenvalues are λ₁ = 5 and λ₂ = 1.
To find the corresponding eigenvectors,
solve the equation (A - λI)v = 0 for each eigenvalue.
Let us start with λ = 5
(A - 5I)v = 0
[tex]|\left[\begin{array}{ccc}1&1\\-5&6\end{array}\right]|[/tex] v = 0
v₁ + v₂ = 0
-5v₁ + v₂ = 0
From the first equation, we get v₂ = -v₁.
Substituting this into the second equation, we have -5v₁ - v₁ = 0,
which simplifies to -6v₁ = 0.
This implies v₁ = 0, and consequently, v₂ = 0.
So, for λ = 5, the eigenvector is v₁ = 0 and v₂ = 0.
Now, let us find the eigenvector for λ = 1.
(A - I)v = 0
[tex]|\left[\begin{array}{ccc}-1&1\\-5&5\end{array}\right][/tex] v = 0
-v₁ + v₂ = 0
-5v₁ + 5v₂ = 0
From the first equation, we get v₂ = v₁.
Substituting this into the second equation, we have -5v₁ + 5v₁ = 0,
which simplifies to 0 = 0.
This implies that v₁ can be any non-zero value.
So, for λ = 1, the eigenvector is v₁ and v₂ = v₁, where v₁ is any non-zero value.
(e) The general solution to the second order ODE can be expressed using the eigenvalues and eigenvectors as follows,
y(x) = c₁ ×[tex]e^{(\lambda_{1} x)[/tex] × v₁ + c₂ × [tex]e^{(\lambda_{2} x)[/tex]× v₂
Plugging in the values we found earlier, the general solution becomes,
y(x) = c₁ × [tex]e^{(5x)[/tex] × [v₁] + c₂× [tex]e^{(x)[/tex]× [v₂]
where [v₁] and [v₂] are the eigenvectors corresponding to the eigenvalues λ₁ = 5 and λ₂ = 1 respectively.
(a) To find the solution to the equation y' - 5y + y = 32,
Use the general solution obtained above.
Comparing the equation with the standard form y' - 5y + 6y = 0,
The equation corresponds to the case where λ₂ = 1.
Substitute λ = 1, v₁ = 1, and v₂ = 1 into the general solution.
y(x) = c₁ × [tex]e^{(5x)[/tex] × [1] + c₂ × [tex]e^{(x)[/tex] × [1]
Simplifying this expression, we have,
y(x) = c₁ × [tex]e^{(5x)[/tex] + c₂ × [tex]e^{(x)[/tex]
Learn more about matrix here
brainly.com/question/31976942
#SPJ4
Luis's car used 4/5
of a gallon to travel 29 miles. At what rate does the car use gas, in miles per gallon?
Answer:
36.25 miles per gallon
Step-by-step explanation:
4/5 = .8
29/.8 = x/1
cross-multiply:
.8x = 29
x = 36.25
Prove that for any x e R, if x2 + 7x < 0, then x < 0. X E
To prove that for any real number x, if x²+ 7x < 0, then x < 0, we can use the properties of quadratic functions and inequalities.
By analyzing the quadratic expression, we can determine the conditions under which it is negative. This analysis shows that the inequality x²+ 7x < 0 holds true when x is less than 0. Consider the quadratic expression x² + 7x. To determine when this expression is negative, we can factor it as x(x + 7). According to the zero product property, this expression is equal to zero when either x or (x + 7) is equal to zero. Thus, the two critical points are x = 0 and x = -7.
Now, let's analyze the behavior of the quadratic expression in the intervals (-∞, -7), (-7, 0), and (0, +∞). Choose a test point from each interval, such as -8, -3, and 1, respectively. Evaluating the expression x²⁺7x for these test points, we find that for -8 and -3, the expression is positive, and for 1, it is positive as well.
Learn more about quadratic expression click here:
brainly.com/question/10025464
#SPJ11
what is the domain of the function
Answer:
hi sh Sheet sh I monorailg Jericho improve Odom Ybor
What is the value of the expression below when x=10x=10?
6x-5
6x−5
Answer:
55
Step-by-step explanation:
To find this, simply plug 10 in for x.
6(10)-5
6*10=60
60-5=55
complete the table... plz help
Answer:
3=20
4=15
5=12
Step-by-step explanation:
for what positive values of k does the function y=sin(kt) satisfy the differential equation y′′ 64y=0?
The function y = sin(kt) satisfies the differential equation y'' - 64y = 0 for pospositiveypospositiveyitiveitive values of k that are multiples of 8.
To determine the values of k for which the function y = sin(kt) satisfies the given differential equation, we need to substitute y into the equation and solve for k. Let's start by finding the first and second derivatives of y with respect to t.
The first derivative of y with respect to t is y' = kcos(kt), and the second derivative is y'' = -k^2sin(kt). Substituting these derivatives into the differential equation gives us:
(-k^2sin(kt)) - 64sin(kt) = 0Simplifying the equation, we get:
sin(kt) = -64*sin(kt)/k^2
We can divide both sides of the equation by sin(kt) (assuming sin(kt) is not zero) to get:
1 = -64/k^2
Solving for k^2, we find k^2 = -64. Since k must be positive, there are no positive values of k that satisfy this equation. Therefore, there are no positive values of k for which the function y = sin(kt) satisfies the given differential equation y'' - 64y = 0.
learn more about differential equation here
https://brainly.com/question/32538700
#SPJ11
hey guys could yall solve this problem for me? thanks
Answer:
Given ABCD ~ EFGH
FG = BC(EF/AB)
FG = 7(9/6)
FG = 63/6
FG = 10.5
GH = CD(EF/AB)
GH = 11(9/6)
GH = 99/6
GH = 16.5
EH = AD(EF/AB)
EH = 12(9/6)
EH = 108/6
EH = 18
The average speed of an airplane is 550 miles per hour Create an equation to represent the distance, d, in miles, that the airplane travels after t hours at the average speed.
Answer:
Step-by-step explanation:
1
Answer:
56
Step-by-step explanation:
i just is smart
yhhhhhhhhhhh
Is this statement true or false? You calculate a finance charge by subtracting the cost of the purchase from the total payment,
Answer:
True
Step-by-step explanation:
Brainliest?
Answer:
I Believe the answer is True
Step-by-step explanation:
just #16 and please show your work!! i will give brainliest!!
Answer:
the bottom triangle is scaled down version of above one
then
[tex] \frac{18}{6} = \frac{x}{2} [/tex]
[tex]x = 6[/tex]
and
[tex] \frac{18}{6} = \frac{y}{5} [/tex]
[tex]y = 15[/tex]
[tex](2x5y6)(4x - 3y - 3) [/tex]
Please please help this is overdue
Answer
only the graph is a function. I can tell by doing the vertical line test on the graph and making sure that each x value only happens once. On the table it is not a function since the x values repeat.Which 3-dimensional figure is associated with the volume formula V = 1/3 π r 2 h?
A. pyramid
B. cylinder
C. sphere
D. cone
Help ASAP!
Answer
c
Step-by-step explanation:
It is not d because of the way that it is formed
what is 1/12 in simplest form
It cant be written any way else its already in its simplest form
Knowledge and Understanding 14. Simplify (1112 - 6vw - 3wa)-(-702 + vw + 13w). 15. Which of the following is equivalent to the expression (5a + 26 - 4c)? a. 25a2 + 20ab - 40ac +482 - 16bc + 1602 b. 25a2 + 10ab - 20ac + 482 - 86C + 16c2 + c. 25a2 + 482 + 1602 d. 10a + 4b-8c 16. Expand and simplify. (b + b)(4 - 5)(25 - 8) 17. Simplify. P-2 3p + 3 X 9p +9 P + 2 3r2 - 18. Simplify. 63 62 po* + 5m3 - 15r + 12 2m2 + 2r - 40 19. Simplify. xi21 4 X + 2 3 x-1
14. (1112 - 6vw - 3wa)-(-702 + vw + 13w) = 1814 - 7vw - 3wa - 13w
15. The equivalent of the expression (5a + 26 - 4c) is 25a2 + 10ab - 20ac + 482 - 86c + 1602 + c.
16. (b + b)(4 - 5)(25 - 8) = -34
14. Simplify (1112 - 6vw - 3wa)-(-702 + vw + 13w).
Given expression is (1112 - 6vw - 3wa)-(-702 + vw + 13w)
⇒ 1112 - 6vw - 3wa + 702 - vw - 13w
⇒ 1814 - 7vw - 3wa - 13w
15. We are to find the equivalent of the expression (5a + 26 - 4c).
a. 25a2 + 20ab - 40ac +482 - 16bc + 1602
b. 25a2 + 10ab - 20ac + 482 - 86C + 1602
c. 25a2 + 482 + 1602
d. 10a + 4b-8c5a + 26 - 4c
= 5a - 4c + 26 = 25a2 - 20ac +482 - 4c2 + 52 - 8ac
= 25a2 - 20ac + 482 - 4c2 + 10a - 8c = Option (b)
⇒ 25a2 + 10ab - 20ac + 482 - 86c + 16c2 + c.
16. Expand and simplify. (b + b)(4 - 5)(25 - 8)
Given expression is (b + b)(4 - 5)(25 - 8) = 2b(-1)(17) = -34
To learn more about simplification
https://brainly.com/question/28008382
#SPJ11
Question 8 of 9
Carlita has a swimming pool in her backyard that is rectangular with a length of 26 feet and a width of 16
feet. She wants to install a concrete walkway of width c around the pool. Surrounding the walkway, she
wants to have a wood deck that extends w feet on all sides. Find an expression for the perimeter of the wood
deck.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
We have given a rectangular swimming pool with a length of 26 feet and a width of 16 feet. We need to find the perimeter of the wood deck that surrounds the concrete walkway of width c around the pool and extends w feet on all sides.
Let's solve the given problem as follows:Firstly, let's calculate the dimensions of the concrete walkway. Let the width of the concrete walkway be 'c' feet.
Then, the width of the pool covered by the concrete walkway is 16 + 2c feet (2c feet on each side), and the length of the pool covered by the concrete walkway is 26 + 2c feet (2c feet on each end).
So, the dimensions of the pool and concrete walkway are (26 + 2c) ft. x (16 + 2c) ft.The dimensions of the wood deck that surrounds the concrete walkway by w feet on all sides will be (26 + 2c + 2w) ft. x (16 + 2c + 2w) ft.Now, let's write the expression for the perimeter of the wood deck.P = 2(Length + Width)P = 2[(26 + 2c + 2w) + (16 + 2c + 2w)]P = 2[42 + 4c + 4w]P = 84 + 8c + 8wThe expression for the perimeter of the wood deck is 84 + 8c + 8w. Hence, the answer is 84 + 8c + 8w.
For more such questions on pool
https://brainly.com/question/28063924
#SPJ8
Three companies, A, B and C, make computer hard drives. The proportion of hard drives that fail within one year is 0.001 for company 0.002 for company B and 0.005 for company C. A computer manufacturer gets 50% of their hard drives from company A, 30% from company B and 20% from company C. The computer manufacturer installs one hard drive into each computer.
(a) What is the probability that a randomly chosen computer purchased from this manufacturer will experience a hard drive failure within one year? [4 marks]
(b) I buy a computer that does experience a hard drive failure within one year. What is the probability that the hard drive was manufactured by company C? [4 marks]
(c) The computer manufacturer sends me a replacement computer, whose hard drive also fails within one year. What is the probability that the hard drives in the original and replacement computers were manufactured by the same company? [You may assume that the computers are produced independently.] [6 marks]
(d) A colleague of mine buys a computer that does not experience a hard drive failure within one year. Calculate the probability that this hard drive was manufactured by company C. [6 marks]
(a) The probability of a computer failure from Company A is 0.001; from Company B is 0.002; and from Company C is 0.005.
Therefore, the probability that a computer will experience a hard drive failure within one year is:(0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005)= 0.0012. The probability of a randomly selected computer experiencing a hard drive failure within one year is 0.0012 or 0.12%.
(b) Bayes' theorem will be used to calculate this probability:Let A be the event that the computer's hard drive was manufactured by Company C. Let B be the event that the computer experienced a hard drive failure. P(A|B) is the probability that the hard drive was manufactured by Company C given that a hard drive failure was experienced.
P(A|B) = P(B|A) P(A) / P(B) Where: P(B|A) = 0.005 (the probability of failure if the hard drive was manufactured by Company C)P(A) = 0.20 (the proportion of hard drives that the computer manufacturer gets from Company C)P(B) = (0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005) = 0.0012 (as in part a)
Therefore: P(A|B) = (0.005 x 0.20) / 0.0012 = 0.0833 or 8.33%.
(c)Let A be the event that both hard drives were manufactured by Company A; B be the event that both hard drives were manufactured by Company B; and C be the event that both hard drives were manufactured by Company C. Then we need to find the probability of event A or B or C, given that a hard drive failure was experienced:P(A U B U C|F) = P(F|A U B U C) P(A U B U C) / P(F)where F is the event that the hard drive in the replacement computer fails.P(F|A U B U C) = P(F) = (0.50 x 0.001) + (0.30 x 0.002) + (0.20 x 0.005) = 0.0012P(A U B U C) = (0.50)^2 + (0.30)^2 + (0.20)^2 = 0.46P(F) = P(A U B U C) P(F|A U B U C) + P(A' n B n C) P(F|A' n B n C)= 0.46 x 0.0012 + 0.04 x 0.3 = 0.000552P(A U B U C|F) = P(F|A U B U C) P(A U B U C) / P(F)= (0.0012 x 0.46) / 0.000552 = 1.00or 100%. Therefore, the probability that the original and replacement computers were produced by the same company is 100%.
(d) Bayes' theorem will be used to calculate this probability:Let A be the event that the hard drive was manufactured by Company C. Let B be the event that the computer did not experience a hard drive failure. P(A|B) is the probability that the hard drive was manufactured by Company C given that no hard drive failure was experienced.P(A|B) = P(B|A) P(A) / P(B)Where:P(B|A) = 1 - 0.005 = 0.995 (the probability that the hard drive did not fail if it was manufactured by Company C)P(A) = 0.20 (as in part b)P(B) = 1 - (0.50 x 0.001) - (0.30 x 0.002) - (0.20 x 0.005) = 0.9988
Therefore:P(A|B) = (0.995 x 0.20) / 0.9988 = 0.1989 or 19.89%. Therefore, the probability that the hard drive was manufactured by Company C given that it did not fail is 19.89%.
#SPJ11
The probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year is approximately 0.256.
(a)Probability that a randomly chosen computer purchased from this manufacturer will experience a hard drive failure within one year = 0.5 x 0.001 + 0.3 x 0.002 + 0.2 x 0.005 = 0.0016
(b)Let's denote the event that a computer failure is experienced within one year by F and the event that the hard drive is made by company C by C.
Then we are required to calculate P(C | F), which is the probability that the hard drive was manufactured by company C given that a failure was experienced by the computer within one year. This can be found by using the Bayes' rule as follows:
[tex]$$P(C|F) = \frac{P(F|C)P(C)}{P(F|A)P(A) + P(F|B)P(B) + P(F|C)P(C)}$$[/tex]
where P(C) = 0.2, P(A) = 0.5 and P(B) = 0.3.$$P(F|A) = 0.001, P(F|B) = 0.002, P(F|C) = 0.005$$
Thus, we have:[tex]$$P(C|F) = \frac{0.005 \times 0.2}{0.001 \times 0.5 + 0.002 \times 0.3 + 0.005 \times 0.2} \approx 0.476$$[/tex]
Therefore, the probability that the hard drive was manufactured by company C given that a failure was experienced by the computer within one year is approximately 0.476.
(c)Let's denote the event that the original hard drive is manufactured by company A, B and C by A, B, and C respectively.
Similarly, let's denote the event that the replacement hard drive is manufactured by company A, B, and C by A', B', and C' respectively.
We are required to calculate P(A = A', B = B', C = C' | F), which is the probability that the hard drives in the original and replacement computers were manufactured by the same company given that a failure was experienced by both computers within one year.
This can be found by using the Bayes' rule as follows:
[tex]$$P(A = A', B = B', C = C'|F) = \frac{P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C')}{P(F)}$$[/tex]
where: [tex]$$P(F) = P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C') + P(F|A \ne A', B \ne B', C \ne C')P(A \ne A')P(B \ne B')P(C \ne C')$$[/tex]
Here, we are assuming that the probabilities of computer failure are independent of each other and the company that manufactured the hard drives of the two computers are independent of each other. Therefore, we have:
[tex]$$P(F|A = A', B = B', C = C') = P(F|A)P(F|B)P(F|C) = 0.001 \times 0.002 \times 0.005$$[/tex]
[tex]$$P(F|A \ne A', B \ne B', C \ne C') = 0$$[/tex]
Also, we have:$$P(A = A') = P(B = B') = P(C = C') = \frac{1}{3}$$
[tex]$$P(A \ne A', B \ne B', C \ne C') = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}$$[/tex]
Thus, we have:$$P(A = A', B = B', C = C'|F) = \frac{0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3}{P(F)}$$
[tex]$$P(A \ne A', B \ne B', C \ne C'|F) = \frac{P(F) - 0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3}{\frac{8}{27}}$$[/tex]
Now, we need to find P(F). This can be done as follows:
[tex]$$P(F) = P(F|A = A', B = B', C = C')P(A = A')P(B = B')P(C = C') + P(F|A \ne A', B \ne B', C \ne C')P(A \ne A')P(B \ne B')P(C \ne C')$$$$= 0.001 \times 0.002 \times 0.005 \times (\frac{1}{3})^3 + 0 = 4.6296 \times 10^{-8}$$Thus, we have:$$P(A = A', B = B', C = C'|F) = 0.0296$$[/tex]
[tex]$$P(A \ne A', B \ne B', C \ne C'|F) = 0.9704$$[/tex]
Therefore, the probability that the hard drives in the original and replacement computers were manufactured by the same company given that a failure was experienced by both computers within one year is 0.0296.(d)Let's denote the event that the hard drive is made by company C by C and the event that a computer failure is not experienced within one year by F'. We are required to calculate P(C | F'), which is the probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year. This can be found by using the Bayes' rule as follows:
[tex]$$P(C|F') = \frac{P(F'|C)P(C)}{P(F'|A)P(A) + P(F'|B)P(B) + P(F'|C)P(C)}$$[/tex]
where P(C) = 0.2, P(A) = 0.5 and P(B) = 0.3.$$P(F'|A) = 0.999, P(F'|B) = 0.998, P(F'|C) = 0.995$$
Thus, we have: [tex]$$P(C|F') = \frac{0.995 \times 0.2}{0.999 \times 0.5 + 0.998 \times 0.3 + 0.995 \times 0.2} \approx 0.256$$[/tex]
Therefore, the probability that the hard drive was manufactured by company C given that a failure was not experienced by the computer within one year is approximately 0.256.
To know more about probability, visit:
https://brainly.com/question/31828911
#SPJ11
How many roots does the equation 3x² = 1 - 7x have and what is the nature of the roots
Answer:
Step-by-step explanation:
Rewrite this quadratic in standard form: 3x^2 + 7x - 1.
The coefficients of x are {3, 7, -1}, and so the discriminant is b^2 - 4ac, or
7^2 - 4(3)(-1), or 49 + 12, or 61. Because the discriminant is positive, this quadratic has two real, unequal roots
instantaneous rate of change for the function:
f(x)= 5x^lnx ; x=3
Use the formula for instantaneous rate of change, approximating the limit by using smaller and smaller values of h, to find the instantaneous rate of change for the given function at the given value. f(x) = 5x x=3 The instantaneous rate of change for the function at x=3 is (Do not round until the final answer. Then round to four decimal places as needed)
The instantaneous rate of change for the function f(x) at x = 3 is 16.9068.
To determine the instantaneous rate of change for the function; f(x) = 5x^ln(x), where x = 3, we can use the formula for the instantaneous rate of change, approximating the limit by using smaller and smaller values of h.
The instantaneous rate of change of the function f(x) at x = 3 can be found as follows:
Let h be a small increment of x that approaches zero. Then the formula for the instantaneous rate of change is given by:
f'(3) = lim[h→0] {(5(3+h)^(ln(3+h))-5(3^(ln3)))/h}
For the above formula, we have: Let f(x) = 5x^ln(x)
Then, f'(x) = 5x^ln(x) * [(d/dx) ln(x)] + 5*ln(x)*x^(ln(x) - 1)
Now, for the given problem, we can substitute 3 for x, and solve as follows: f'(3) = 5(3^ln(3)) * [(d/dx) ln(x)] + 5*ln(3)*3^(ln(3) - 1)
f'(3) = 5(3^ln(3)) * [(1/x)] + 5*ln(3)*3^(ln(3) - 1)
f'(3) = 5(3^ln(3)) * [(1/3)] + 5*ln(3)*3^(ln(3) - 1)
f'(3) = 5(3^ln(3) / 3) + 5*ln(3)*3^(ln(3) - 1)
Then, f'(3) = 5e ln(3) + 5 ln(3) / 3= (5e ln(3) + 5 ln(3) / 3)= 16.9068 (rounded to four decimal places).
Therefore, the instantaneous rate of change for the function f(x) at x = 3 is 16.9068.
Know more about instantaneous rate here,
https://brainly.com/question/30760748
#SPJ11