[SOLVED] Probability and statistics (ucs410)

Experiment 4
(Mathematical Expectation, Moments and Functions of Random Variables)1. The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given as
x 0 1 2 3 4
p(x) 0.41 0.37 0.16 0.05 0.01
Find the average number of imperfections per 10 meters of this fabric.
(Try functions sum( ), weighted.mean( ), c(a %*% b) to find expected value/mean.2. The time T, in days, required for the completion of a contracted project is a random variable with probability density function f(t) = 0.1 e(-0.1t) for t > 0 and 0 otherwise. Find the expected value of T.
Use function integrate( ) to find the expected value of continuous random variable T.3. A bookstore purchases three copies of a book at $6.00 each and sells them for $12.00 each. Unsold copies are returned for $2.00 each. Let X = {number of copies sold} and Y = {net revenue}. If the probability mass function of X is
x 0 1 2 3
p(x) 0.1 0.2 0.2 0.5Find the expected value of Y.4. Find the first and second moments about the origin of the random variable X with probability density function f(x) = 0.5e-|x|, 1 < x < 10 and 0 otherwise. Further use the results to find Mean and Variance.
(kth moment = E(Xk), Mean = first moment and Variance = second moment – Mean2.5. Let X be a geometric random variable with probability distributionWrite a function to find the probability distribution of the random variable Y = X2 and find probability of Y for X = 3. Further, use it to find the expected value and variance of Y for X = 1,2,3,4,5.