Chapter 3.1, Problem 21E

Let $X_1,X_2,...,X_k$ be random variables of the continuous type, and left $f_1\left(x\right),f_2\left(x\right),\mathrm{...},f_k\left(x\right)$ be their corresponding pdfs, each with sample space $S=\left(-\infty ,\infty \right)$.

Also, let $c_1,c_2,\mathrm{...},c_k$ be nonnegative constants such that ${\displaystyle {\sum }_{i=1}^k{c}_i=1}$.

(a) Show that ${\displaystyle {\sum }_{i=1}^k{c}_i{f}_i\left(x\right)}$ is a pdf of a continuous-type random variable on S.

(b) If X is a continuous-type random variable with pdf ${\displaystyle {\sum }_{i=1}^k{c}_i{f}_i\left(x\right)}$ on $S,E\left(X_i\right)={\mu }_i$ and $\mathrm{var}\left(X_i\right)={\sigma }_i^2$ for i = 1,..., k, find the mean and the variance of X.