X Y Z iid Suppose for random variable X, P(X > a) - exp( random variable...
Let h be an exponentially-distributed random variable with the distribution function p- exp(-x) for x > 0 and ph = nction Ph 0 for a s 0. Derive the distribution function of its square root, Solution: 2y exp(-y2
Let > 0 and a > 0 be given. Suppose that X is a random variable with moment generating function e My(t) = {(A-ta tsy Top til Compute Var(X). Show that if we define Ly(t) = In My(t) then Ls (0) = Var(X).
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2.5.1. The probability function of a random variable Y is given by p (i)-% 1 0, 1, 2, , where λ is a known positive value and c is a constant. (a) Find c. (b) Find P(Y 0) (c) Find P(Y > 2).
The cumulative distribution function of the random variable X is given by F(x) = 1-e-r' (z > 0). Evaluate a) P(X > 2) b) P(l < X < 3 c) P(-1 〈 X <-3). d) P(-1< X <3)
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 be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X > 28]. Answer: CHECK
I. (5 points) Let X be a random variable with moment generating function M(t) = E [etx]. For t > 0 and a 〉 0, prove that and consequently, P(X > a inf etaM(t). t>0 These bounds are known as Chernoff's bounds. (Hint: Define Z etX and use Markov inequality.)
5. A non-negative valued continuous random variable X satisfies P(X > x +y|X > x) = P(X > y) > 0 for any x,y > 0. (a) Show that P(X > nx) = [P(X > x)]" and P(X > x/m) = [P(X > x)]1/m for positive integers n, m. (b) Show that X~ exponential() for some A > 0.
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint: Determine first the probability distribution of T -X +Y + Z using the moment generating function method. Moment generating function for Poisson random variable is given in earlier lecture notes)
Let X, Y and Z be three independent Poisson random variable with parameters λι, λ2, and λ3, respectively. For y 0,1,2,t, calculate P(Y yX+Y+Z-t) (Hint:...
1. Given a continuous random number x, with the probability density P(x) = A exp(-2x) for all x > 0, find the value of A and the probability that x > 1.