If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?

Answers

Answer 1

The distribution of Z = X/Y is  λ₁ / (λ₁+λ₂).

What is a cumulative distribution function?

The probability distribution of random variables is described using the cumulative distribution function. The probability for a discrete, continuous, or mixed variable may be described using it. The cumulative probability for a random variable is calculated by adding the probability density function.

Given:

X and Y are independent exponential random variables with respective parameters λ₁ and λ₂.

To find the distribution of Z = X/Y:

First, we have

[tex]f_x_y[/tex](x,y) = λ₁λ₂[tex]e^{-\lambda_1x[/tex][tex]e^{-\lambda_2y[/tex]

First, we find the cumulative distribution function (CDF) for Z = X/Y.

Derivative of Z, f(z) and put  a = x/y

[tex]F_z[/tex](a)  = P (X/Y ≤ a)

        = P (X ≤ aY)

        = [tex]\int\limits^{\infty}_ {x} \ \int\limits^{ay}_0 {\lambda_1\lambda_2e^{-\lambda_1x}e^{-\lambda_2y}\ dx dy[/tex]

       [tex]= \int\limits^{\infty}_ {0} {\lambda_1\lambda_2e^{-\lambda_2y}\ dy [ -1/{\lambda_1}{e^{-\lambda_1x}]\limits^{ay}_ {0}[/tex]

       [tex]= \lambda_2\int\limits^{\infty}_ {0} {e^{-\lambda_2y} - e^{-y(\lambda_2 + \lambda_1a)} \ dy[/tex]

        [tex]= \lambda_2[ [{1/{\lambda_2}+{\lambda_1a]-{e^{-\lambda_2y} + e^{-y(\lambda_2 + \lambda_1a)}]\limits^{\infty}_ {0}[/tex]

        [tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1) - 1][/tex]

        [tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1a) - 1][/tex]

       [tex]= [({\lambda_1a}/{\lambda_2 + \lambda_1a)][/tex]

So, P(X<Y) = P (X/Y <1)

                 = λ₁ / (λ₁+λ₂)

Therefore, distribution is λ₁ / (λ₁+λ₂).

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Related Questions

An object travels along a horizontal straight path at a constant rate. The object travels 1/2 of the
length of the path in 3/4 second. At that rate, how many seconds does it take the object to travel the
entire length of the path?

Answers

Answer:

1.5 seconds

Step-by-step explanation:

Given that,

The object travels 1/2 of the  length of the path in 3/4 second.

We need to find the time it takes the object to travel the entire length of the path.

1/2 of length = 3/4 seconds

Entire length,

[tex]=\dfrac{3}{4}\times 2\\\\=1.5\ s[/tex]

So, it will take 1.5 seconds to cover the entire length of the path.

What is the answer for (a^2+12c)÷7b-1

Answers

Answer:

Step-by-step explanation:

Integer roots:

a = 6 n + 1, b = 7, c = -3 n^2 - n, n element Z

a = 6 n + 5, b = 7, c = -3 n^2 - 5 n - 2, n element Z

Derivative:

d/da(1/7 (a^2 + 12 c) b - 1) = (2 a b)/7

Indefinite integral:

integral(-1 + 1/7 b (a^2 + 12 c)) da = (a^3 b)/21 + (12 a b c)/7 - a + constant

Related Queries:

What is the better deal? 6 for $6.99 or 24 for $13.99

Answers

6.99/6 =1.16
13.99/24=.58
Therefore better to buy more as the unit price per item is cheaper

Answer:

24 for $13.99

Step-by-step explanation:

Find the cost of 1 for both deals.

6 for $6.99. Divide 6.99 with 6:

6.99/6 = 1.165

~$1.17 for one.

24 for $13.99. Divide 13.99 with 24:

13.99/24 = ~0.582

~$0.58 for one.

24 for $13.99 is the better deal, by ~59¢.

IF THERe ARE 8 Cats, 12 dogs
and 4 RABBITS THEN
ratio 1;3 IS
FOR?

Answers

Answer:

1:3 ratio is for cats

Step-by-step explanation:

ratio of cats: no. of cats/ no. of rabbit+ cats+dogs

= 8/8+12+4

= 1/3

= 1:3

Hahaha Kauai’s rabis mosni 1:3-7

-27 +2+ 23р - Зр
Hurry please

Answers

Answer:

20p - 25

Step-by-step explanation:

-27 +2+ 23р - Зр

20p - 25

Answer:

-25+20p

Step-by-step explanation:

Add -27 and 2

-25+23p-3p

Subtract 3p from 23 p

-25+20p

Hope this helps! If so brainiest, please?

Thanks!!!

the length and width of a rectangle are represented by a 2x+3 and x-1 respectively, which of the following represents the perimeter of the rectangle.

Answers

Its very simple just apply the rule then apply the equations he gave you i hope u understand it n goodluck

Allison collected books to donate to different charities. The following expression can be used to determine the number of books each charity received.



(12 + 4 • 5) ÷ 2



Based on this expression, how many books did each charity receive?

Answers

The answer is 16


Explanation

(12+4•5)divided by 2

(12+20)divided by 2

32 divided by 2

=16

Look at the photo for an Easy question and get easy points

Answers

Answer:

90

Step-by-step explanation:

why my answer has to be 20 characters long? :)

Answer:

the answer is 90

Step-by-step explanation:

Hope I helped!

Task: Volume & Surface Area
Instructions
The following question is a two part question. In the first part, you will be
asked to solve for volume. In the second part, you will be asked to solve for
surface area
Complete each of the 2 activities for this Task.
Activity 1 of 2
A storage chest is shown. What is the volume in cubic feet?
2 feet
2 feet
4.5 feet
Activity 2 of 2
A storage chest is shown. What is the surface area in square feet?
2 feet
2 feet
4.5 feet
help me please!!! someone help me out please!!!!!!

Answers

Okay so starting off I get that this math can be really confusing but you’ll get it if you keep working at it. Don’t give up you got this. For part it it is asking for the volume in cubic feet to do this you multiple H•W•L

H=hight

W=Width

L=Length

Now you don’t really need to know exactly which side is the H,L, or W but since you only have 3 sides with domentions you know that you multiply all three sides now it doesn’t matter what order you do it in.

Please respond if this helped for part A. You got this keep your head up.

Cassie paid $276.45 for 5 months of cell phone service. At that rate, how much will Cassie pay for a full year of cell phone service?

NO ANSWERS FROM UNKNOWN FILES PLEASE OR I'LL REPORT IT!!!

Answers

It’s 663.48 because if you divide 276.45 by 5 than you get 55.19 and multiply that by 12 and it’s 663.48

Plz, give me an answer thanks :)

Answers

Answer:

D

Step-by-step explanation:

A research study examined the blood vitamin D levels of the entire US population of landscape gardeners. The population average level of vitamin D in US landscapers was found to be 30.8 ng/mL with a standard deviation of 4.371 ng/mL. Assuming the true distribution of blood vitamin D levels follows a normal distribution, if you randomly select a landscaper in the US, what is the likelihood that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

Answers

Answer:

0.0631 = 6.31% probability that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

Step-by-step explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

The population average level of vitamin D in US landscapers was found to be 30.8 ng/mL with a standard deviation of 4.371 ng/mL

This means that [tex]\mu = 30.8, \sigma = 4.371[/tex]

What is the likelihood that his/her vitamin D level will be between 36.84 and 39.73 ng/mL?

This is the pvalue of Z when X = 39.73 subtracted by the pvalue of Z when X = 36.84.

X = 39.73

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{39.73 - 30.8}{4.371}[/tex]

[tex]Z = 2.04[/tex]

[tex]Z = 2.04[/tex] has a pvalue of 0.9793

X = 36.84

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{36.84 - 30.8}{4.371}[/tex]

[tex]Z = 1.38[/tex]

[tex]Z = 1.38[/tex] has a pvalue of 0.9162

0.9793 - 0.9162 = 0.0631

0.0631 = 6.31% probability that his/her vitamin D level will be between 36.84 and 39.73 ng/mL

Which of the following equations has
infinitely many solutions?
A. 2x + 3 = 5 + 2x
B. 2x + 3 = 5 + 3x
C. 3x - 5 = -5 + 2x
D. 3x - 5 = -5 + 3x

Answers

The One With Inifinetely Many Solutions is D. 3x-5=-5+3x are both the same.Meaning that no matter Which Value u Put For X u will get the same Answer On Both Side
D
Hope this helps you

the triangles shown are similar. the side measuring 4 in. corresponds to the side measuring 9 in. what is the ratio ( smaller to larger )

Answers

Answer:

[tex]Ratio = \frac{9}{4}[/tex]

Step-by-step explanation:

Given

[tex]Small =4in[/tex]

[tex]Large = 9in[/tex]

Required

The scale ratio

From small to big, the scale ratio is:

[tex]Ratio = Big : Small[/tex]

[tex]Ratio = 9in:4in[/tex]

Cancel out the units

[tex]Ratio = 9:4[/tex]

Express as fraction

[tex]Ratio = \frac{9}{4}[/tex]

What is the slope and y-intercept of this line?
y=3x + 7

Answers

Answer:

the slope is 3 and the y-intercept is 7

Step-by-step explanation:

in slope-intercept form, y=mx+b, m = the slope, and b = the y-intercept

Hello there!

The equation of this line is written in slope-intercept form, which looks like so:

y=mx+b

Where

m is the slope (-1/2 for this line)

b is the y-intercept (7)

Hope it helps.

~A lonely teen who helps others and listens to "How Far I'll Go

-Grace ;-)

Good luck.

A board game has two spinners. The first spinner has 4 equal areas labeled 1, 2, 3 and 4. The second
spinner has 3 equal areas that are labeled: Move Ahead, Move Back, and Do Not Move. What is the
probability the first spinner will land on a 2 and the second spinner will land. on Move Back? Pls help I’m begging u pls I need this pls help have a good day pls help

F.1/2
G.1/4
H.1/8
J.1/12

Answers

Answer:

J

Step-by-step explanation:

prob of landing 4 x prob of landing move back

1/4 x 1/3 = 1/12

Hunter has a rectangular prism with a volume of 1,155 cubic cm. He measures the length of the prism as 5 1/2 cm and the width of the prism as 10 1/2 cm.
The height of Hunter’s rectangular prism must be

Answers

Answer:

John is buying shirts for his softball team.  He will pay a one-time processing fee of $27.50 and $12.75 per each shirt ordered.  Which fffadasdequation can be used to find y, the total cost to buy xshirts? John is buying shirts for his softball team.  He will pay a one-time processing fee of $27.50 and $12.75 per each shirt ordered.  Which equation can be used to find y, the total cost to buy xshirts? John is buying shirts for his softball team.  He will pay a one-time processing fee of $27.50 and $12.75 per each shirt ordered.  Which equation can be used to find y, the total cost to buy xshirts? vv

Step-by-step explanation:

A store randomly samples 602 shoppers over the course of a year and finds that 144 of them made their visit because of a coupon they'd received in the mail. Construct a 95% confidence interval for the proportion of all shoppers during the year whose visit was because of a coupon they'd received in the mail. (Round your answers to three decimal places.)

Answers

Answer:

well as a percent its 23.92

Step-by-step explanation:

What is 4.9 subtract 0.6

Answers

Answer: 4.3

Step-by-step explanation:

4.9-.6=4.3

A student purchased 9 binders for $11.07.

Use the drop down to answer the question.

Which equation can be used to find the cost of each binder, b, in dollars?

Answers

Answer:

11.07/9

Step-by-step explanation:

Math

A circle has a radius of 4. An arc in this circle has a central angle of 288º.
What is the length of the arc?

Answers

Answer:

its  [tex]\frac{32\pi }{5}[/tex] or in degree it is 1152º

Step-by-step explanation:

Figure A ~ Figure B
Figure A
Figure B
1 ft
3 ft
V = 6 cu ft
Find the volume of Figure A.

Answers

Answer:

the answer is

162ft3

Step-by-step explanation:


In the equation 5(x2 + 6x) + 3 (y2 - 2y) = 77, complete the square on x by adding within the first parentheses. Complete the square on y by
ning Guide
adding within the second parentheses. Thus, add to the right side of the equation.
In the equation 5(x2 + 6x) + 3(y2 - 2y) = 77, complete the square on x by adding ?????what within the first parentheses. Complete the square on y by adding ????? What

within the second parentheses. Thus, add to the right side of the equation.

Answers

Answer:

a=5

Step-by-step explanation:

Answer:

7                                                                                                   8

Step-by-step explanation:

DUE TOMORROW HELP!!!!!!

Shannon is planning to make a model of a pyramid for
a geography project. The length of each edge of the
base is 11 in., and the height of each triangular face
8.4 in. How much cardboard will Shannon need to
make the model pyramid?

Answers

Answer:

c

Step-by-step explanation:

edge

The answer to your question is C

The doctor tells Emily that she needs to exercise enough to burn at least 200 calories each day. She prefers kickboxing or swimming. While kickboxing she burns 10 calories per minute, and while swimming she burns 7 calories per minute.

Answers

Answer: 10x+7y _>

200

Step-by-step explanation:

The best-fit equation is 10x+7y _> 200

What are fit equations?

Fitting an equation to data is finding a linear, quadratic, exponential, or any other sort of function whose graph includes, or comes as close as possible to, a given set of data in the form of ordered pairs. The relative predictive power of a model shows how accurately the model is related to the data.

The theoretical pattern best accounts for the relationships among variables in a data set. For example, a regression equation having the best fit to sample data is the one that minimizes differences between observed and predicted values.

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The number of three digit integers less than 501 have exactly two different digits in the presentation for (example 232 or 466) is___.

Thanks in advance:)​

Answers

Answer:

73

Step-by-step explanation:

Five chicken nuggets have 180 calories. How many calories are in 7 chicken nuggets?
0 36 calories
0 252 calories
0 900 calories
O 1, 260 calories

Answers

Answer:

The correct answers 252 calories

Step-by-step explanation:

How I figured it out was by dividing 180 by 5, which gave me 36. Then 36(7) which equaled out to be 252.

elliaarteaga14
hey ik u dont know me,but i found u and i need a lot of help. Please and thank you If Mrs. Allan’s car uses 8 gallons of gas for a 224-mile trip, then, at that rate, how many gallons of gas would the car use for a 560-mile trip?

Answers

Answer:20 gallons

Step-by-step explanation:

Sketch the graph of y=-2x² + x + 1 using your graphing calculator. What are the x-intercepts of this graph?
a. (1, 0) and (-0.5, 0)
c. There are no x-intercepts
b. (-2.5, 0) and (-2,0)
d. (-1.5, 0) and (-0.5, 0)
Please select the best answer from the choices provided
A
B
0
D

Answers

Answer is going to be A.
The answer to your question is A.

PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA 40 POINTS* DONT SKIP :(( .!

Answers

Answer:

VERTICAL ANGLES

Step-by-step explanation:

because since it’s right on the opposite of each other they would be vertical cuz they are the same angle measurement.

GOOD LUCK MY BROTHA

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