Let X and Y be independent variables with X ~ EXP(mu(x)) and Y ~ EXP (mu(y)), where mu(x) = 1 and mu(y) = 1/2. Write explicit integral expressions for each of the following, without computing the values. P(Y < X)
Let X and Y be independent variables with X ~ EXP(mu(x)) and Y ~ EXP (mu(y)),...
Let X and Y be independent random variables with X = N(0, 1) and Y = Exp(1). Find E( |X| (Y + 1)^2 ).
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.
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 11. (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? (e) Compute PIZ> t where t20. (f) Compute the pd.f. of Z. Z = min(X,Y)
Let X and Y be independent exponential random variables with pa- rameter ? = 1. Given that X and Y are independent, their joint pdf is given by the product of the individual pdfs of X and Y , that is, fX,Y(x, y) = fX(x) fY(y). The joint pdf is defined over the same set of x-values and y-values that the individual pdfs were defined for. Using this information, calculate P (X ? Y ? 2) where you can assume...
Question 5 - Even More Fun With Bivariate Normal Distributions Let X and Y be independent normally distributed with mean x = 2 and μΥ--3 and standard deviations ơX-3 and ơY-5, respectively. Determine the following: (a) P(3X 6Y>15), (b) P(3X6Y<30) (c) Cov(X, Y) d) Verify (a) and (b) using R code, where for each case you generate a million X's and a million Y's and simulate the linear combination 3X 6Y. (e) Assume now that the random variables come from...
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)...
4. Let X and Y be independent exponential random variables with pa- rameter ? 1. Given that X and Y are independent, their joint pdf is given by the product of the individual pdfs of X and Y, that is, fxy(x,y) = fx(x)fy(y) The joint pdf is defined over the same set of r-values and y-values that the individual pdfs were defined for. Using this information, calculate P(X - Y < t) where you can assume t is a positive...
Let X and Y be two independent random variables with X =d R(0, 2) and Y =d exp(1). (a) Use the convolution formula to calculate the probability density function of W =X+Y. (b) Derive the probability density function of U = XY .
7. Suppose that Xi,..., Xk are independent random variables, and X, ~ Exp(B) for i = 1, . . . , k. Let Y = min(X1 , . . . , Xk). Show that Y ~ Exp(Σ-1 β).
(15 points) Consider two independent, exponential random variables X,Y ~ exp(1). Let U = X + Y and V = X/(X+Y). (a) (5 points) Calculate the joint pdf of U and V. (b) (5 points) Identify the distribution of U. If it has a "named” distribution, you must state it. Otherwise support and pdf is enough. (c) (5 points) Identify the distribution of V.If it has a "named” distribution, you must state it. Otherwise support and pdf is enough.