Question

a) If X1 and X2 are independent random variables and X1 tollows the Nor nalLA σ1 X, +X2 follow? di tri t on and X to ows the Nonna μα 2 distribution, ne ha distribution do b) IfX1 , X2 . X, , arendependent random variables and each Xk follows the NormalA 에 ds rbutio. then what distribution does follow? , n L.6) Generating functions for sums of independent random variables a) If X and X are independent random variables, then how is the generating function of Xi X2 related to the generating function of X1 and the generating function of X? b) If X1 , X2 X3 .. X are independent random variables, then how do you get the generating function of Xi +X2+X3+...X from the generating functions of XiX^ .Xs ....X-i .and X?

Answer 1

a)

We know that the linear combination of Normal random variables follow Normal distribution.

E[X1 + X2] = E[X1] + E[X2] = $\mu_1 + \mu_2$

Var[X1 + X2] = Var[X1] + Var[X2] = $\sigma_1^2 + \sigma_2^2$

Standard deviation of X1 + X2 is $\sqrt{\sigma_1^2 + \sigma_2^2}$

Thus,

X1 + X2 follows the Normal[$\mu_1 + \mu_2$, $\sqrt{\sigma_1^2 + \sigma_2^2}$ ]

b)

We know that the linear combination of Normal random variables follow Normal distribution.

$E\left [ \sum_{k=1}^{n}X_k \right ] = \sum_{k=1}^{n}E[X_k] \right = \sum_{k=1}^{n} \mu_k \right$

$Var\left [ \sum_{k=1}^{n}X_k \right ] = \sum_{k=1}^{n}Var[X_k] \right = \sum_{k=1}^{n} \sigma^2_k \right$ (Since all Xk are independent random variables)

Standard deviation of $\sum_{k=1}^{n}X_k$ is,

$\sqrt{\sum_{k=1}^{n} \sigma^2_k \right}$

Thus, $\sum_{k=1}^{n}X_k$ follows the

$Normal\left [ ~\sum_{k=1}^{n} \mu_k , \sqrt{\sum_{k=1}^{n} \sigma^2_k} \right]$

a)

Moment generating function of X1 + X2 is product of moment generating function of X1 and X2.

$M_{X_1+X_2} (t)= M_{X_1} (t).M_{X_2} (t)$

b)

Moment generating function of X1 + X2 + ... + Xn is product of moment generating function of X1, X2, ..., Xn-1 , Xn.

$M_{X_1+X_2+ ..+ X_n} (t)= M_{X_1} (t).M_{X_2} (t)...M_{X_{n-1}} (t).M_{X_n} (t)$