Mean and Variance of Y using Expectation Operator

Question

Let

$$ Y =\sum_{k=1}^N a_kX_k $$

be the weighted sum of N independent random variables, $ X_k, k = 1, ... , N $ , each having mean $ \mu _{X_i} $ and variance $ \sigma ^2_{X_i} $. The weights $a_k$ are real-valued constants.

• Derive an expression for the mean of $ Y, \mu _Y $
• Derive an expression for the variance of $Y, \sigma ^2_Y $

Hint: Use the expectation operator...

Attempt:

$E[Y] = \mu _Y$ , or the expectation of $Y$ is the mean of $Y$. Since $Y$ represents the weighted sum of the N independent random variables, dividing $Y$ by the summation of the weights should give the mean, or:

$$ \mu _Y = E[Y] = Y / \sum_{k=1}^N a_k $$

where $Y$ is defined above.

Then we have $\sigma ^2_Y = E[Y^2] - \mu ^2_Y $ which can be derived from the above but gets very messy. Am I on the right track?

Answer 1

The expectation is a linear operator, so:

$$E[Y]=E\Biggl[\sum_{i=1}^Na_iX_i\Biggr]=\sum_{i=1}^Na_iE[X_i]=\sum_{i=1}^Na_i\mu_{X_i}$$

Then, you already know that $$V[Y]=E[Y^2]-E^2[Y]$$

So $$\begin{array}{rcl} E[Y^2]&=&E\Biggl[\Biggl(\sum_{i=1}^Na_iX_i\Biggr)^2\Biggr]\\ &=&E\Biggl[\sum_{i=1}^Na_i^2X_i^2+2\sum_{1\leq i<j\leq N}a_ia_jX_iX_j\Biggr]\\ &=&\sum_{i=1}^Na_i^2E\Bigl[X_i^2\Bigr]+2\sum_{1\leq i<j\leq N}a_ia_jE\Bigl[X_iX_j\Bigr]\\ \end{array}$$

You know that $E[X_i^2]=E^2[X_i]+V[X_i]=\mu_{X_i}^2+\sigma_{X_i}^2$, and $X_i$ are independent, so: $$\begin{array}{rcl} E[Y^2]&=&\sum_{i=1}^Na_i^2(\mu_{X_i}^2+\sigma_{X_i}^2)+2\sum_{1\leq i<j\leq N}a_ia_j\mu_{X_i}\mu_{X_j}\\ \end{array}$$

Answer 2

It might be easier if you give yourself an example. Say each week you collect cans and bottles to recycle. Each can yields $0.05$ cents and each bottle yields $0.10$ cents. Let $X_1$ denote the amount of cans you find each week and $X_2$ the amount of bottles; on average you find $\mu_{X_1}$ cans and $\mu_{X_2}$ bottles. What's the average amount of money you make each week? Answer:

\begin{gather*} 0.05 \times \mu_{X_1} + 0.10 \times \mu_{X_2}. \end{gather*}

For the second part, note that \begin{gather*} E\left[\left(\sum_{i=1}^N X_i\right)^2\right] = E\left[\sum_{i=1}^N X_i^2 + 2\sum_{1 \leq i < j \leq N} X_i X_j\right]. \end{gather*} If $X_i$ and $X_j$ are independent, then $E[X_iX_j] = E[X_i]E[X_j]$.

Answer 3

$$ \mu_y=\sum_{i=1}^k a_k E(X_k)=\mu \sum_{i=1}^k a_ix_i $$ Since $X_i$ are independent : $$ \sigma_y^2=\sum_{i=1}^k a_k^2 VAR(X_k)=\sigma^2 \sum_{i=1}^k a_i^2 x_i $$ EDIT Since $\mu$ and $\sigma$ have now the subscripts they should go under the sums. $$ \mu_y=\sum_{i=1}^k a_k E(X_k)=\sum_{i=1}^k a_i\mu_i $$ $$ \sigma_y^2=\sum_{i=1}^k a_k^2 VAR(X_k)=\sum_{i=1}^k a_i^2 \sigma^2_i $$