1. Let X be a random variable having pdf f(x) = 6x(1 – x) for 0 < I < 1 and 0 elsewhere.Compute the mean and variance of X.2. Let X1, X2,..., X, be independent random variables having the same distribution as the variablefrom problem 1, and let X, = (X1+ ·.+Xn).Part a: Compute the mean and variance of X, (your answer will depend on n).Part b: If I didn't assume the variables were independent, would the calculation in part a stillwork? Or would at least part of it still work?3. Suppose that X and Y are both independent variables, and that each has mean 2 and variance3. Compute the mean and variance of XY (for the variance, you may want to start by computingE(X²Y²)).4. Suppose that (X,Y) is a point which is equally likely to be any of {(0, 1), (3,0), (6, 1), (3, 2)}(meaning, for example, that P(X = 0 and Y = 1) = }).Part a: Show that E(XY) = E(X)E(Y).Part b: Are X and Y independent? Explain.5. Let X be a random variable having a pdf given byS(2) = 2e-2" for 0

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1. Let X be a random variable having pdf f(r) = 6r(1 – 2) for 0 < z < 1 and 0 elsewhere. Compute the mean and variance of X.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

1. Let X be a random variable having pdf f(x) = 6x(1 – x) for 0 < I < 1 and 0 elsewhere.
Compute the mean and variance of X.
2. Let X1, X2,..., X, be independent random variables having the same distribution as the variable
from problem 1, and let X, = (X1+ ·.+Xn).
Part a: Compute the mean and variance of X, (your answer will depend on n).
Part b: If I didn't assume the variables were independent, would the calculation in part a still
work? Or would at least part of it still work?
3. Suppose that X and Y are both independent variables, and that each has mean 2 and variance
3. Compute the mean and variance of XY (for the variance, you may want to start by computing
E(X²Y²)).
4. Suppose that (X,Y) is a point which is equally likely to be any of {(0, 1), (3,0), (6, 1), (3, 2)}
(meaning, for example, that P(X = 0 and Y = 1) = }).
Part a: Show that E(XY) = E(X)E(Y).
Part b: Are X and Y independent? Explain.
5. Let X be a random variable having a pdf given by
S(2) = 2e-2" for 0 <I< ∞, 0 otherwise .
Part a: Compute the moment generating function of X. For what t is it defined?
Part b: Using your answer to part a, determine E(X), E(X²), and Var(X).

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