Means and Variance of Linear Combinations of Random Variables

• Some useful properties that will simplify the calculations of means and variances of random variables.
• These properties will permit us to deal with expectations in terms of other parameters that are either known or are easily computed.
• Theorem 4.5:
  
• Corollary 4.1: $E(b) = b$
• Corollary 4.2: $E(aX) = aE(X)$

• Example 4.16: Applying Theorem 4.5 to the continuous random variable $g(X) = 4X + 3$, the density function of $X$ is as follows.
  $$f(x)=\left\lbrace \begin{array}{l} \frac{x^2}{3}~ for~ -1 < x < 2 \\ 0, ~elsewhere \\ \end{array}\right\rbrace$$

• Solution:
  $$E(4X+3)=4E(X)+3=4\left( \int_{-1}^2 x\frac{x^2}{3}dx\right)+3=8$$

• Theorem 4.6:
  

• Theorem 4.7:
  
• Corollary 4.3: Setting $g(X, Y) = g(X)$ and $h(X, Y) = h(Y)$.
  $$E[g(X)\pm h(Y)]=E[g(X)] \pm E[h(Y)]$$

• Corollary 4.4: Setting $g(X, Y) = X$ and $h(X, Y) = Y$.
  $$E[X\pm Y]=E(X) \pm E(Y)$$

• Theorem 4.7:
  
• Corollary 4.5: Let $X$ and $Y$ be two independent random variables, Then $\sigma_{XY}=0$
  • $E(XY) = E(X)E(Y)$ for independent variables
  • $\sigma_{XY}=E(XY)-E(X)E(Y)=0$

• Example 4.19: In producing gallium-arsenide microchips, it is known that the ratio between gallium and arsenide is independent of producing a high percentage of workable wafers.
• Let $X$ denote the ratio of gallium to arsenide and $Y$ denote the percentage of workable wafers retrieved during a 1-hour period.
• $X$ and $Y$ are independent random variables with the joint density being known as
  $$f(x)=\left\lbrace \begin{array}{l} \frac{x(1+3y^2)}{4}  fo... ...,   0 < y < 1 \\ 0, elsewhere \\ \end{array}\right\rbrace$$
  Illustrate that $E(XY) = E(X)E(Y)$.

• Solution:
  $$E(XY)=\int_0^1 \int_0^2 xyf(x,y)dxdy=\int_0^1 \int_0^2 xy\frac{x(1+3y^2)}{4}dxdy=\frac{5}{6}$$
  $$E(X)=\int_0^1 \int_0^2 xf(x,y)dxdy=\int_0^1 \int_0^2 x\frac{x(1+3y^2)}{4}dxdy=\frac{4}{3}$$
  $$E(Y)=\int_0^1 \int_0^2 yf(x,y)dxdy=\int_0^1 \int_0^2 y\frac{x(1+3y^2)}{4}dxdy=\frac{5}{8}$$
  $$(E(XY) =)\frac{5}{6}=\frac{4}{3}* \frac{5}{8}(=E(X)*E(Y))$$

• Theorem 4.9:
  
• Corollary 4.6: $\sigma_{X+b}^2 = \sigma^2_X=\sigma^2$
  • The variance is unchanged if a constant is added to or subtracted from a random variable.
  • The addition or subtraction of a constant simply shifts the values of $X$ to the right/left but does not change their variability.
• Corollary 4.7: $\sigma_{aX}^2 = a^2\sigma^2_X=a^2\sigma^2$
  • The variance is multiplied or divided by the square of the constant.

• Theorem 4.10:
  
• Corollary 4.8: If $X$ and $Y$ are independent random variables, then
  $$\sigma_{aX+bY}^2 = a^2\sigma_X^2+b^2\sigma_Y^2$$

• Corollary 4.9: If $X$ and $Y$ are independent random variables, then
  $$\sigma_{aX-bY}^2 = a^2\sigma_X^2+b^2\sigma_Y^2$$

• Corollary 4.10: If $X_1,X_2, \ldots X_n$ are independent random variables, then
  $$\sigma_{a_1X_1+a_2X_2+\ldots a_nX_n}^2 = a_1^2\sigma_{X_1}^2+a_2^2\sigma_{X_2}^2+\ldots +a_n^2\sigma_{X_n}^2$$

• Example 4.20: $X$ and $Y$ are random variables with variances $\sigma_X^2=2$, $\sigma_Y^2=2$, and covariance $\sigma_{XY}=-2$,
• Find the variance of the random variable $Z = 3X - 4Y + 8$
• Solution:
  $$\sigma_Z^2=\sigma_{3X - 4Y + 8}^2=\sigma_{3X - 4Y}^2 (by Theorem 4.9)$$
  $$=9\sigma_X^2+16\sigma_Y^2-24\sigma_{XY}~(by~Theorem~4.10)$$
  $$=130$$

• Example 4.21: Let $X$ and $Y$ denote the amount of two different types of impurities in a batch of a certain chemical product.
• Suppose that $X$ and $Y$ are independent random variables with variances $\sigma_X^2=2$, $\sigma_Y^2=3$
• Find the variance of the random variable $Z = 3X - 2Y + 5$
• Solution:
  $$\sigma_Z^2=\sigma_{3X - 2Y + 5}^2=\sigma_{3X - 2Y}^2 (by Theorem 4.9)$$
  $$=9\sigma_X^2+4\sigma_Y^2 (by Corollary 4.9)$$
  $$=30$$