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Let ? and ? be independent random variables. Random variable ? has mean ?? and variance...

Let ? and ? be independent random variables. Random variable ? has mean ?? and variance ?^2?, and random variable ? has mean ?? and variance ?^2?
a) Prove that ?[?⋅?]=??⋅??
Guidance: Start with ?[?⋅?]=ΣΣ??⋅???(?,?)??, and then use the definition of independent random variables.
b) Use a) to prove that ???(??+??)=?^2???(?)+?^2???(?).
Guidance: Use the formula proved in the class ???(?)=?[?^2]−?^2[?].
c) Let ? =5?+3?. Find the mean and variance of ? in terms of the means and variances of ? and ?.

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