Suppose X and Y are independent random variables with Exponential(2) distribution (Section 6.3). We say X...
2. If X and Y are independent random variables, X has a normal distribution with mean 2 variance 4, and Y has a chi-square distribution with 9 degrees of freedom, then find u such that P(X > 2+11,7)=0.01.
2. Suppose X and Y are independent continuous random variables. Show that P(Y < X) = | Fy(x) · fx (x) dx -oo where Fy is the CDF of Y and fx is the PDF of X [hint: P[Y E A] = S.P(Y E A|X = x) · fx(x) dx]. Rewrite the above equation as an expectation of a function of X, i.e. P(Y < X) = Ex[•]. Use the above relation to compute P[Y < X] if X~Exp (2)...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
Random variables z and y described by the PDF if x-+ yo 1 and x.> 0 and y, > 0 0 otherwise a Are x and y independent random variables? b Are they conditionally independent given max(x,y) S 0.5? c Determine the expected value of random variabler, defined byr xy.
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
1 x Suppose X has an exponential distribution, thus its pdf is given by fx (x) = 5e8,0 5x<0, 2> 0;0 0.w. a. Find E(X) b. Find E(X(X-1) c. Find Var (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.
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.
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;
(6 points) Let X and Y be independent random variables with p.d.f.s fx(x) -{ { 1-22 0, for |2|<1, otherwise. fy(y) = for y>0, otherwise. 0, Let W = XY (a) (2 points) Find the p.d.f. of W, fw(w). (b) (2 points) Find the moment generating function of W2, Mw?(t) = E (e«w?). (c) (2 points) Find the conditional expectation of W given Y = y, E(W|Y = y).