Let  X be a continuous random variable with range  [1,3]  and its probability density function is given by the following function: f(x)=c/(x^3) where  c is a constant.  (1) Find the value of  c . Answer:   (2) Find the expected value of  X . Answer:

Let X be a continuous random variable whose probability density function is: f(x)- 1.5x2 for -1

Let X be a random variable with probability density function (PDF) given by, find E(e²x) a b C d -G fx(x) = x < 0 otherwise

Let X be a random variable with a probability density function given by for-1 < x < 2; otherwise. Let Y = X². £x (x) = { 0, ², (a) Find the probability density function of Y using the distribution function method. (b) Find the probability density function of Y using the change of variable method.

Let X be a random variable with probability density function [ a + bx², 0

Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 otherwise. Q2(i.) Sketch(Shade) the support of (X1, X₂). Q2 (ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2(iii.) Let Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint…

Q1) CDF, PDF, Expectation and Variance The cumulative distribution function (CDF) of random variable Y is y 1. a) Find the probability density function, fy(y), of random variable Y. b) Show that f fy (y)dy = 1. c) What is E[Y]? d) What is VAR[Y]?

Let X be a random variable with the probability density function f(x) = { ae; x>0 0 ; otherwise Find (i) Value of a

The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.