Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and...
2. Let Xand Y be random variables with joint moment generating function M(s,t) 0.3+0.1es + 0.4e +0.2 es*t (a) What are E(X) and E(Y)? (b) Find Cov(X,Y)
2. Let Xand Y be random variables with joint moment generating function M(s,t) 0.3+0.1es + 0.4e +0.2 es*t (a) What are E(X) and E(Y)? (b) Find Cov(X,Y)
5) Let X be a random variable with density Find the moment generating function. State the values of t for which the moment generating function exists.
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.)
Question 4. [5 marksi Let Xbe a random variable with probability mass function (pmf) A-p for -1, 2,... and zero elsewhere (whereq-1-p, 0 <p< (a) Find the moment generating function (mg ofX. C11 (b) Using the result in (a) or otherwise find the expected value and variance of X. C23 (c) Let X, X,., X, be independent random variables all with the pmf fix) above, and let Find the mgf and the cumulant generating function of Y.
Let be a random variable with probability density function f(x) and moment-generating function 1 1 M(t) = =+ = ? 6 . 6 1 + - 1 36 + -e a) Calculate the mean = E(X) of X b) Calculate the variance o? = E(X -w' and the standard deviation of X
Let X be a discrete random variable. If the moment generating function of X is given by (1 -0.9+0.9e) 15. The first moment of X is Hint: Write the answer with one decimal point. Answer.
Let X be a discrete random variable. If the moment generating function of X is given by (1 – 0.6 + 0.6e')? The first moment of X is 8 Hint: Write the answer with one decimal point. Answer:
Let X be a continuous random variable with values in [ 0, 1], uniform density function fX(x) ≡ 1 and moment generating function g(t) = (e t − 1)/t. Find in terms of g(t) the moment generating function for (a) −X. (b) 1 + X. (c) 3X. (d) aX + b.
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Problem 2 Prove the following bound known as the Chemoff bound: Let X be a random variable with moment generating function X (s) defined for s > 0, Then for any a and any s > 0, Hint: To prove the bound apply Markov's inequality with X replaced by e) Apply the се Chemoff bound in case X is a standard normal random variable and a > 0. Find the value of s >0 that gives the sharpest bound, i.e,...