Let X and Y have the joint probability distribution as following Y 0 1 3 a 0.2 0.1 0.3 0.2 0.1 (1) Find a; (2) Get the marginal probability distributions of (X,Y) (3) Determine if X and Y are independent or not 1.
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- Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.IF F(x, y) is the value of the joint distribution function of X and Y at (x, y), show that the marginal distribution function of X is given by G(x) = F(x, ∞) for - ∞ <x < ∞ Use this result to find the marginal distribution function of X for the random variable F(x, y) = { (1-e-x2 ) (1- e-y2) for x>0, y> 0 and 0 elsewhereLet X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]
- Let the random variables X and Y have a joint PDF which is uniform over the triangle with verticies at (0,0),(0,1), and (1,0). Find the joint PDF of X and Y Find the marginal PDF of Y FInd the condtional PDF of X given Y Find E[X[Y=y], and use the total expectation theorem to find E[X] in terms of E[Y] Use the symmetry of the problem to find the value of E[X]5. Determine the value of c that makes the function f (x, y) = c(x + y) a joint probability density function over the range 0 < x < 3 and 0 < y < x. Answer is C = 2/27 a. From the answer in Question 5, Determine the marginal probability distribution of X. b. From the answer in Question 5, Determine the conditional probability distribution of X given that Y = 2. c. From the answer in Question 5, Determine E(Y | X = 1). (up to 6 decimal place)7)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: X~ f (x I θ) . Find the distribution function of the random variable X.
- let X be a binomial random variable with parameters n and θ1. Conditional on X = m, let Y be a binomial random variable with parametersm and θ2.(a) What is the marginal distribution of Y ?(b) Find the functions g(k) = E(Y |X = k) and v(k) = V ar(Y |X = k).Use these to find E(Y ) and V ar(Y ).(c) Find Covariance(X, Y ) and Correlation(X, Y ).Discrete random variables X and Y have a joint distribution function F_X, Y (x, y) = 0.10 u(x + 4) u(y - 1) + 0.15u(x + 3)u(y + 5) + 0.17u(x + 1)u(y - 3) + 0.05 u(x) (y - 1) + 0.18u(x - 2) u(y + 2) + 0.23u(x - 3)u(y - 4) + 0.12u(x - 4)u(y + 3) Find: (a) the marginal distributions F_X(x) and F_Y(y) and sketch the two functions, (b) X and Y, and (c) the probability P{-1 < X lessthanorequalto 4, -3 < Y lessthanorequalto 3}.Let X have a probability density function A) Let Y = X1 + X2 + … + X25 be the sum of a random sample of size 25 from the distribution whose pdf is f(x). Approximate P[12 < Y < 21]. B) Let Yn = X1 + X2 + … + Xn be the sum of a random sample of size n from the distribution whose pdf is f(x). Discuss lim n->infinity P[-14.4 < Yn < 28.8]
- 1)Let x and y be two continuous random variables whose function is the probability density joint is given by : a)Draw the relationship between the variables x and y on the Cartesian axes .b)Calculate the marginal pdfs px(X)and py(Y).c)Are the v.a.s x and y independent ?1. A continuous random variable X has a probability density function (PDF) p(x) = k( 8-x^2/2) on the interval [0, 4] (a) Find k such that p(x) is a valid PDF. (b) Find P(X ≤ 1). (c) Find the mean, µ, of X. (d) Find the variance, σ 2 , of X.Two discrete random variables X and Y have the joint probability density function: f(x.y)=e*p" (1-p)* y!(x-y)! x-y ,y 0,1,2,..,x; x 0,1,2,... where 2, p are constants with 2 >0 and 0 <p<1. 1. Find the marginal probability density functions of X and Y. 2. Find the conditional distribution of Y for a given X and of X for a given Y.
