We have X and Y two independent exponential random variables with E(X) = 1 and E(Y) = 3. Therefore:
and for x > 0, y > 0
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find...
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
2) Let X and Y be independent exponential random variables with means E[X] = 0 and EY = 28. 1 1 f(310) = -X/0 e x > 0, f(y|0) = e-4/20 y > 0 0 24 a) Show that the likelihood function can be written as (2 points) L(0) = e-3(x+3) 202 b) Find the MLE ô of 0. (5 points)
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
. Let X and Y be the proportion of two random variables with joint probability density function f(x, y)o, elsewhere. (a) Find P(X < 3|Y= 2). (b) Are X and Y independent? Why? (c) Find E(Y/X)
Problem 8: 10 points Suppose that (X, Y) are two independent identically distributed random variables with the density function defined as f (x) λ exp (-Ar) , for x > 0. For the ratio, z-y, find the cumulative distribution function and density function.
. Let X and Y be the proportion of two random variables with joint probability density function f(r, y) e-*, 0, if, 0 < y < x < oo, elsewhere. a) Find P(Xc3.y-2). b) Are X and Y independent? Why? c) Find E(Y/X)
Suppose X and Y are independent random variables with Exponential(2) distribution (Section 6.3). We say X ~ Exponential(2) if its pdf is f(x) = -1/2 for x > 0.
t X and Y be independent random variables with variance ơ1 and ơ3. respectively. Consider the sum. Z=aX + (1-a)% 0 < a < 1 Le Find a that minimizes the variance of Z
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.