Problem 10. Show that if X and Y are independent exponential random variables with λ distribution....
1. Let X and Y be two independent exponential random variables with parameters λ and μ, respectively. Compute the probability P(X Y| min(X,Y)-x).
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditional on X> Y, the random variables min(X, Y) and X -Y are independent. (b) Use part (a) to conclude that for any positive constant c E[min(X, Y)IX > Y + c] = E[min(X, Y)|X > Y] = E[min(X, Y)] = λ+p (c) Give a verbal explanation of why min(X, Y) and X - Y are (unconditionally) independent.
33. Let X...
Let X1, X2, ..., Xr be independent exponential random variables with parameter λ. a. Find the moment-generating function of Y = X1 + X2 + ... + Xr. b. What is the distribution of the random variable Y?
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.
6. Let X, Y be independent random variables, each having Exponential(A) distribution. What is the conditional density function of X given that Z =
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
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.
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
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.