Problem 8: 10 points Suppose that (X, Y) are two independent identically distributed random variables with...
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.
4.3. 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.
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common density function f(x) 2x 0〈x〈1 0 otherwise Find the probability density function of X Y. 42.5 If 0 < a < l then ÍxHY(a) 2a3. If 1 < a < 2 then ÍxHY(a) -릎a3 + 4a-3. If a 〉 2 then fx+y(a) 0 and 0 otherwise.
2. Suppose X and Y are independent random variables with the pdf (probability density func- tion) f(x)- for x > 0. (a) What is the joint probability density function of (X, Y)? (b) Define W = X-Y, Z = Y, then what is the joint probability density function fw,z(w, z) for (W, Z). (c) Determine the region for (w, z) where fw,z is positive. (d) Calculate the marginal probability density function for W
2. Suppose X and Y are independent random variables with the pdf (probability density func- tion) f(x) e-2 for x > 0. (a) What is the joint probability density function of (X, Y)? (b) Define W-X-Y, Z = Y, then what is the Joint probability density function fw.z(w, z) for (W, Z). (c) Determine the region for (w, z) where fw.z is positive. (d) Calculate the marginal probability density function for W.
Q1. Assume that X is Pareto random variables with the density -α-1 , r21, where α > 0 (a) Calculate EX]. What do you need to assume about a for E[X to be finite? (b) Find the density of X + b for b 〉 0. (c) Find the cumulative distribution function of Y log X.
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
7. The random variables X and Y have joint probability density function f given by 1 for x > 0, |y| 0 otherwise. 1-x, Below you find a diagram highlighting the (r, y) pairs for which the pdf is 1 (a) Calculate the marginal probability density function fx of X (b) Calculate the marginal cumulative distribution function Fy of Y (c) Are X and Y independent? Explain.
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.