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##### Probability

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**Question**

Question;10. a)Y has moment generating function;mY (z) = e^?8z+32z^2. What is the;variance of Y? What is P(Y??4)?;b)Use the moment generating function to derive the expected value of a;Poisson random variable with parameter? = 1/3.;c)Let Y? Uniform(?,?),?,??;R and???Y}= 1/(1+y);9. (a);The random variable X has moment generating function;mX (t) = exp (?5t + 12t^2).;If Z = 1/5 (X? 2), what are E(Z) and Var(Z)?;?(b) Y is a random variable with moment generating function mY (t) = (1/(1?;2t))^8.;What are E(Y) and Var(Y)?;8.;Let Z1 and Z2 be independent normal random variables where Z1? N(6,3^2) and Z2?;N(4,1^2). Define Y1 = Z1?Z2 and Y2 = 2(Z1) +3(Z2). Find the means and;variances of the random variables Y1 and Y2. [6];The random variable X has moment generating function mX (t) = exp(?7t + 18t^2).;If W = 1/5(X? 2) what are the expected value and variance of W?;10. (a);The random variable X has moment generating function;mx(t)=(1?2t)^?15/2.;If Z = 1/3(X + 3), what is the variance V(Z)?;?(b) Y is a random variable with moment generating function mY (t) = (0.8 +;0.2e^t)^10.;What is P(1?Y?4)?;11. (a);The random variable X takes the values 0, 1, 2 with probabilities 1/2, 3/8;1/8;respectively. Find the moment generating function for X and verify that the;second moment of X is 7/8.;[5];(b);The random variable Z has distribution function FZ (z) = 1? exp(?5z) for z? [0,?). Find the moment generating function for Z and;verify that the first moment of Z is 0.2.

Paper#60397 | Written in 18-Jul-2015

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