(6) Suppose that X is an absolutely continuous random variable with density 1<I<2 f(3) = lo,...
1. (10 points) Let X be a continuous random variable with the probability density function given by f(x)-4z if 0SaS1 and O otherwise (a) Find P(X sjIx> j) (b) Find the expectation and variance of X
Suppose that X and Y are jointly continuous random variables with joint probability density function f(x,y) = {12rºy, 1 0, 0<x<a, 0<y<1 otherwise i) Determine the constant a ii) Find P(0<x<0.5, O Y<0.25) HE) Find the marginal PDFs fex) and y) iv) Find the expected value of X and Y. Le. E(X) and E(Y) v) Are X and Y independent? Justify your answer.
(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.
8. Suppose X is a continuous random variable with density f(x) = 1/3,-1 < x < 0, f(x) = 2(z-1)2,1 < x < 2, and 0 everywhere else. (a) Find E(X).
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 otherwise (a) Determine the moment generating function of Y (b) Suppose the random variables X each have a continuous uniform distribution on [0,1 for i 1,2. Show that the random variable Z X1X2 has the same distribution = as the random variable Y defined above. Consider the random variable Y, whose probability density function is defined...
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)?
please I need detailed explanation The density function of a continuous random variable X is 4(9-2) 8 i 0 elsewhere Find: (a)the mean, mode and median of X. (b) the semi-interquartile range and the mean deviation of the distribution (c) the coefficient of skewness and kurtosis of the distribution.
X with density fcx)3/56 ir 2<<4 5. Consider a continuous random variable X with density f(x)- otherwise a. Find P(1 <X<3) b. Find ECX)
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).