\\[ \\begin{array}{l} \\Rightarrow \\quad=2 x\\left(\\frac{n}{2}+1\ ight) \\\\ \\text { So } \\mu_{2}=\\operatorname{var} \\overrightarrow{(x)}=2 n\\left(\\frac{n}{2}+1\ ight)-n^{2} \\\\ =2 n \\\\ \\end{array} \\] Random Variables IV (November 2023) 1. Suppose \\( M_{X}(t)=\\frac{1}{8} e^{-5 t}+\\frac{1}{4} e^{t}+\\frac{3}{8} e^{7 t} \\). Find a formula for \\( E\\left(X^{n}\ ight) \\). 2. Suppose \\( M_{X}(t)=\\frac{9}{(3-t)^{2}} \\). Find a formula for \\( E\\left(X^{n}\ ight) \\). 3. Suppose \\( M_{X}(t)=e^{\\left(3 e^{t}-3\ ight)} \\). Find \\( \\operatorname{Var}(X) \\). 4. Suppose \\( X \\) is a random variable which takes on the values \\( 1,2,3 \\), and 4 , with 0.3 , and 0.4 , respectively. Find \\( M_{X}(t) \\). 5. Given the moment-generating function \\( M_{X}(t)=e^{3 t+8 t^{2}} \\) of a random variable \\( X \\) generating function of the random variable \\( Z=\\frac{1}{4}(X-3) \\) and use it to find the \\( m \\) of \\( Z \\). Random Variables IV (November 2023) Examples . The discrete random variable \\( X \\) has the pdf \\[ p_{X