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A amogos lf X and Y are independent exponential random variables with parameters 11 and 12...
1. Let X and Y be two independent exponential random variables with parameters λ and μ,...
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).
6. Let X, Y be independent random variables, each having Exponential(A) distribution. What is the conditional...
6. Let X, Y be independent random variables, each having Exponential(A) distribution. What is the conditional density function of X given that Z =
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditio...
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...
12. Let X and Y be independent random variables, where X has a uniform distribution on...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter = 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(x > 0.25) U (Y > 0.25)}? (c) What is the conditional distribution of X. given that Y - 3? (d) What is Var(Y - E[2X] + 3)? (e) What is...
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x >...
Let X and Y be independent exponential random variables with pdfs f(x) = λe-λx (x > 0) and f(y) = µe-µy (y > 0) respectively. (i) Let Z = min(X, Y ). Find f(z), E(Z), and Var(Z). (ii) Let W = max(X, Y ). Find f(w) (it is not an exponential pdf). (iii) Find E(W) (there are two methods - one does not require further integration). (iv) Find Cov(Z,W). (v) Find Var(W).
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter A= 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(X > 0.25) U (Y> 0.25)}? nd (c) What is the conditional distribution of X, given that Y =3? ur worl mple with oumbers vour nal to complet the ovaluato all...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let ...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random...
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]
L.11) Sums of independent random variables a) If X1 , X2 X, , , Xn are...
L.11) Sums of independent random variables a) If X1 , X2 X, , , Xn are independent random variables all with Exponential μ distribution, then what is the distribution of XII + 2 +X3 + .tX b) If X is a random variable with Exponential[u] distribution, then what is the distribution of x +X1? c) If X1 , X2 , Х, , , X are independent random variables all with Normal 0. I distribution, then what is the distribution of...
Problem 10. Show that if X and Y are independent exponential random variables with λ distribution....
Problem 10. Show that if X and Y are independent exponential random variables with λ distribution. Also, identify the degrees of freedom. 1,then X/Y follows an F
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).
6. Let X, Y be independent random variables, each having Exponential(A) distribution. What is the conditional density function of X given that Z =
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...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter = 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(x > 0.25) U (Y > 0.25)}? (c) What is the conditional distribution of X. given that Y - 3? (d) What is Var(Y - E[2X] + 3)? (e) What is...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter A= 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(X > 0.25) U (Y> 0.25)}? nd (c) What is the conditional distribution of X, given that Y =3? ur worl mple with oumbers vour nal to complet the ovaluato all...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]
L.11) Sums of independent random variables a) If X1 , X2 X, , , Xn are independent random variables all with Exponential μ distribution, then what is the distribution of XII + 2 +X3 + .tX b) If X is a random variable with Exponential[u] distribution, then what is the distribution of x +X1? c) If X1 , X2 , Х, , , X are independent random variables all with Normal 0. I distribution, then what is the distribution of...
Problem 10. Show that if X and Y are independent exponential random variables with λ distribution. Also, identify the degrees of freedom. 1,then X/Y follows an F
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