STAT 115 Let X be a continuous random variable having the CDF Fx(x) = 1 -...

Question

Question

STAT 115 Let X be a continuous random variable having the CDF

Fx(x) = 1 - e^ (-e^x)

(1) Find the Probability Density Function (PDF) of Y=e^X.

(2) Let B have a uniform distribution over (0,1). Find a function G(b) and G(B) has the same distribution as X.

math Statistics-And-Probability

Answer 1

Answer #1

1)

P(Y < y)

= P(e^x < y)

= P(X < ln y )

= F(ln y)

= 1 - e^ (-e^(ln y))

= 1- e^(-y))

hence

f(y) = d/dy F(y) = e^(-y)

Y follow exponential distribution with mean = 1

Please give me a thumbs-up if this helps you out. Thank you! :)

By HomeworkLib answering guidelines, we have to answer only one question at a time

Answer 2

Similar Homework Help Questions

STAT 120 Let X be a continuous random variable having the CDF Fx(x) = 1 -...

STAT 120 Let X be a continuous random variable having the CDF Fx(x) = 1 - e^ (-e^x) Let B have a uniform distribution over (0,1). Find a function G(b) and G(B) has the same distribution as X.
Problem # 8. a) Let X be a continuous random variable with known CDF FX(x). LetY = g(X) where g(·) is the so-called sign...

Problem # 8. a) Let X be a continuous random variable with known CDF FX(x). LetY = g(X) where g(·) is the so-called signum function, which extracts the sign of its argument. In other words, g(X) = { -1 x<0, 0 x=0, 1 x>0 } Express the PDF fY (y) in terms of the known CDF FX(x). b) Let X be a random variable with PDF: fX(x) = { x/2 0 <= x < 2, 0 otherwise} Let Y be...
X is a positive continuous random variable with density fX(x). Y = ln(X). Find the cumulative...

X is a positive continuous random variable with density fX(x). Y = ln(X). Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...

1. Let X be a continuous random variable with the probability density function fx(x) = 0...

1. Let X be a continuous random variable with the probability density function fx(x) = 0 35x57, zero elsewhere. Let Y be a Uniform (3, 7) random variable. Suppose that X and Y are independent. Find the probability distribution of W = X+Y.
Question 3: Let X be a continuous random variable with cumulative distribution function FX (x) =...

Question 3: Let X be a continuous random variable with cumulative distribution function FX (x) = P (X ≤ x). Let Y = FX (x). Find the probability density function and the cumulative distribution function of Y . Question 3: Let X be a continuous random variable with cumulative distribution function FX(x) = P(X-x). Let Y = FX (x). Find the probability density function and the cumulative distribution function of Y
3. (10 points) Let X be a continuous random variable with CDF for x < -1...

3. (10 points) Let X be a continuous random variable with CDF for x < -1 Fx(x) = { } (x3 +1) for -1<x<1 for x > 1 and let Y = X5 a. (4 points) Find the CDF of Y. b. (3 points) Find the PDF of Y. c. (3 points) Find E[Y]

Proble 2. Let Fx(t) be the cumulative distribution function (CDF) of a continuous random variable X...

Proble 2. Let Fx(t) be the cumulative distribution function (CDF) of a continuous random variable X and let Y-X. Express the CDF of Y terms of Fx(t).
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random var...

Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find...

(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).

1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a....

1. Let X and Y be two jointly continuous random variables with joint CDF otherwsie a. Find the joint pdf fxy(x, y), marginal pdf (fx(x) and fy()) and cdf (Fx(x) and Fy)) b. Find the conditional pdf fxiy Cr ly c. Find the probability P(X < Y = y) d. Are X and Y independent?