Y1 A) Write PDF function of X? fx (x)- B) Find P(YSO.5)? C) Find Varl)
For a continuous random variable X with the following probability density function (PDF): fX(x) = ( 0.25 if 0 ≤ x ≤ 4, 0 otherwise. (a) Sketch-out the function and confirm it’s a valid PDF. (5 points) (b) Find the CDF of X and sketch it out. (5 points) (c) Find P [ 0.5 < X ≤ 1.5 ]. (5 points)
For a random variable X with cumulative distribution function (cdf) Fx(x) = 1- (2/x)^2 ,x>2. (a).Find the pdf fX(x). (b).Consider the random variable Y = X^2. Find the pdf of Y, fY (y).
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)?...
Suppose X is an exponential random variable with PDF, fx(x) exp(-x)u(x). Find a transformation, Y g(X) so that the new random variable Y has a Cauchy PDF given 1/π . Hint: Use the results of Exercise 4.44. ) Suppose a random variable has some PDF given by ). Find a function g(x) such that Y g(x) is a uniform random variable over the interval (0, 1). Next, suppose that X is a uniform random variable. Find a function g(x) such...
5. Let fx(x) be a pdf given by fx(x) = (1/8)(e^(-x/8)) for x > 0. a) Find the CDF FX(x). b) Find P(X > 4) c) Find P(-2 ≤ X ≤ 12) d) Find P(X < 240) e) Find E(X) f) Find the standard deviation of X.
Suppose the random variable X has PDF fX(x) = 3x2 when 0 < x < 1 and zero otherwise. What is the PDF of Y = 2X+3? a.Use the general method: Find the CDF of X and use it to get the PDF of Y b.Use a short-cut.
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().) 2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
2. A continuous random variable X has PDF SPI? 1€ (-2,2] fx() = 0 otherwise (a) Find the CDF Fx (x). (b) Suppose 2 =9(X), where gle) = { " Find the (DF, PDF of
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
5. (20 pts) Function of RV Let Ry X-Exponential(1),i.e.,the CDF is Fx (x) = (1 - )u(x). IEX = 9(x) = -2x + 1, find the CDF Fy (y) and the PDF fy(y).