Ans. For a given continuous distribution, the distribution of any transformation is obtained by using the Jacobian of transformation.

4) Suppose that Y~exp(8). Let X = ln(Y). Find the pdf of X. 5) Let Y...
7) Suppose that Y1 and Y2 are iid exp(8). Find the pdf of R-½/
Let X and Y be iid uniform random variables on [0,1]. Find the pdf of Z=X+Y
6. Suppose X and Y have the joint pdf fr,y) = 2 exp(-:- 0 ) 0< <y otherwise o a. Find Px.x, the correlation coefficient between X and Y. b. Let U = 2X-1 and V=Y +2. What is pu.v, the correlation coefficient between U and V? c. Repeat (b) if U = -TX and V = Y + In 2. d. Let W = Y - X. Compute Var (W). e. Refer to (d). Find an interval that will...
a) Let X-Unif(0,1). Derive the pdf of Y =-ln(1-X) Remember to provide its support. Let X-N(1,02). Derive the pdf of Y-ex and remember to provide its support. b) Hint for both parts: First work out the cdf of Y, and then use it to find the density of Y.
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
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...
Let X have the pdf defined for 0<x<2. Let Y~Unif(0,1). Suppose X and Y are independent. Find the distribution of X-Y. fx() =
6. (10 points) Suppose X ~ Exp(1) and Y = -ln(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y.
Let Xi, , X. .., Exp(β) be IID. Let Y max(Xi, , h} Find the probability density function of Y. İlint: Y < y if and only if XS for i 1,,n.
Matlab question: using the following example: generate an exponentially distributed RV y, with a exponential pdf with a parameter a=0.3 i) plot the histogram (using 1 million points) ii) generate a data-driven histogram to see the goodness of mach example: suppose: pdf: fy(y)=a exp(-ay)u(y) cdf: Fy(y) = [1-exp(-ay)]u(y) transform uniform RV to a exponential RV: Y=F^-1y=-ln(1-x)/a but because x is a uniform distributed (0,1) than y = -ln(x)/a) thanks