1. Show that the sum of independent and identically distributed Poisson random variables is Poisson distributed....

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1. Show that the sum of independent and identically distributed Poisson random variables is Poisson distributed. 2. Verify the formula for the first two moments of a gamma distributed random variable. 3. Consider a two-dimensional normally distributed random vector (X,Y) ~ N(, ) with EY = My, Var(X)=on, Var(Y)=o2, Cov(X,Y)=012. EX = x, Show that 022 E(Y|X) = f(c) = ky+ 012 (2 uc ) = Uy t (x x), 011 011 012 is the correlation coefficient. 011022 where g = 4. Let X,..., X100 be independent, identically distributed (1,4) random variables and Find P(S 140). 100 S=>X. i=1 1. Show that the sum of independent and identically distributed Poisson random variables is Poisson distributed. 2. Verify the formula for the first two moments of a gamma distributed random variable. 3. Consider a two-dimensional normally distributed random vector (X,Y) ~ N(, ) with EY = My, Var(X)=on, Var(Y)=o2, Cov(X,Y)=012. EX = x, Show that 022 E(Y|X) = f(c) = ky+ 012 (2 uc ) = Uy t (x x), 011 011 012 is the correlation coefficient. 011022 where g = 4. Let X,..., X100 be independent, identically distributed (1,4) random variables and Find P(S 140). 100 S=>X. i=1