3.81 The random variable X has moment generating function M (1) = 0.2e41 + 0.7e7t +...
Exercise 1 Let X be a random variable that has moment generating function My(t) = 0.5-t2-t Find P[-1<x< 1]
(3 marks) The moment generating function of a random variable X is given by MX(t) = 24 20 < - In 0.6. Find the mean and standard deviation of X using its moment generating function.
+ 2 A continuous random variable Y has moment generating function m(t) = e50t+251-72. Find (a) P(40 <Y < 45) (b) a value b such that P(Y < b) = 0.975.
The geometric random variable X has moment generating function given by EetX) = p(1 – qe*)-7, where q = 1- p and 0 < p < 1. Use this to derive the mean and variance of X.
(1 point) If X is a random variable with moment generating function ui) = (1-1)-9, t < I/7 then E(X) = and Var(X) =
2. A random variable has a probability density function given by: Bmx-(B+1) x20 x<m fx(x)= 10 where m>0 and B > 2. Let m and ß be constants; answer the questions in terms of m and B. (a) Find the cumulative distribution function (cdf) Fx(x) of this random variable; (b) Find the mean of X; (c) Find E[X']; and (d) Find the variance of X. [12 points]
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
I. (5 points) Let X be a random variable with moment generating function M(t) = E [etx]. For t > 0 and a 〉 0, prove that and consequently, P(X > a inf etaM(t). t>0 These bounds are known as Chernoff's bounds. (Hint: Define Z etX and use Markov inequality.)
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
2-2.3 A probability distribution function for a random variable has the form F,(x) = A(1-exp[-(x-1)) 1 < x < oo -00<xs1 a) For what value of A is this a valid probability distribution function? b) What is Fx (2)? c) What is the probability that the random variable lies in the interval 2 X < 00? d) What is the probability that the random variable lies in the interval 1 <X s3?