1. Suppose the random variable X has the following probability density function:
1. Suppose the random variable X has the following probability density function: Problem Set: 1. Suppose...
Problem #2: Suppose that a random variable X has the following probability density function. SC(16 - x?) 0<x< 4 f(x) = 3 otherwise Find the expected value of x.
Let X be a random variable with the following probability density function: 0 otherwise. Using following relationship ueudu a. Show that fy (y) is a valid probability density function b. Show that the moment generating function My (t) =-for t 2 (2-t) c. Obtain the first and second raw moments. d. Using these raw moments determine the mean and variance Let X be a random variable with the following probability density function: 0 otherwise. Using following relationship ueudu a. Show...
Consider the random variable X whose probability density function is given by k/x3 if x>r fx(x) = otherwise Suppose that r=5.2. Find the value of k that makes fx(x) a valid probability density function.
(1) Suppose that X is a continuous random variable with probability density function 0<x< 1 f() = (3-X)/4 i<< <3 10 otherwise (a) Compute the mean and variance of X. (b) Compute P(X <3/2). (c) Find the first quartile (25th percentile) for the distribution.
I . (20%) Random variable X has the probability density function as ; Random variable Y 2X+1 0 otherwise a) Determine A b) Determine the Probability Distribution Function F, (x) c) Determine E(X) and ơx d) Determine the probability density function fy(y) and E(Y)
4. (30pts) A continuous random variable X has the probability density function: hx - 1 sx 32 f(x) =Jo-hx 2 x 3 0 x >3 which ean bo graphed as f(x) 1 2 a) Find h which makes f(x) a valid probability density function b) Find the expected value E(X) of the probability density function f(x) c) Find the cumulative distribution function F(x). Show all you work
Consider a continuous random variable X with the following probability density function: Problem 2 (15 minutes) Consider a continuous random variable X with the following probability density function: f(x) = {& Otherwise ?' 10 otherwise? a. Is /(x) a well defined probability density function? b. What is the mathematical expectation of U (2) = x (the mean of X, )? c. What is the mathematical expectation of U(z) = (1 - 2 (the variance of X, oº)?
2. Let U be a continuous random variable with the following probability density function: 1+1 -1 <t<o g(t) = { 1-1 03151 0 otherwise a. Verify that g(t) is indeed a probability density function. [5] b. Compute the expected value, E(U), and variance, V(U), of U. (10)
4 (3 points) Suppose a random variable X has the following probability density function: 3x2 -1srs0 0 otherwise f(x) (a) Compute Pr[Xs-1/2 (b) Compute E (X), the expectation of x (c) Compute the cumulative distribution function of this random variable (for all real numbers).
1. 20 points Let X be a random variable with the following probability density function: f(x)--e+1" with ? > 0, ? > 0, constants x > ?, (a) 5 points Find the value of constant c that makes f(x) a valid probability mass function. (b) 5 points Find the cumulative distribution function (CDF) of X.