12.4 Problems on Variance, Covariance, Linear Regression

(See Exercise 4 from "Problems on Mathematical Expectation"). In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of each lightning strike starting a fire is about 0.0083. Determine $Var \left[X\right]$.

(See Exercise 5 from "Problems on Mathematical Expectation"). Two coins are flipped twenty times. Let X be the number of matches (both heads or both tails). Determine $Var \left[X\right]$.

(See Exercise 6 from "Problems on Mathematical Expectation"). A residential College plans to raise money by selling “chances” on a board. Fifty chances are sold. A player pays $10 to play; he or she wins $30 with probability $p = 0 . 2$. The profit to the College is

Determine $Var \left[X\right]$.

(See Exercise 7 from "Problems on Mathematical Expectation"). The number of noise pulses arriving on a power circuit in an hour is a random quantity X having Poisson (7) distribution. Determine $Var \left[X\right]$.

(See Exercise 24 from "Problems on Distribution and Density Functions", and Exercise 8 from "Problems on Mathematical Expectation"). The total operating time for the units in Exercise 24 from "Problems on Distribution and Density Functions" is a random variable $T \sim$ gamma (20, 0.0002). Determine $Var \left[T\right]$.

The class $\left\{A , B , C , D , E , F\right\}$ is independent, with respective probabilities

0.43, 0.53, 0.46, 0.37, 0.45, 0.39. Let

1. Use properties of expectation and variance to obtain $E \left[X\right]$, $Var \left[X\right]$, $E \left[Y\right]$, and $Var \left[Y\right]$. Note that it is not necessary to obtain the distributions for X or Y.
2. Let $Z = 3 Y - 2 X$.
   Determine $E \left[Z\right]$, and $Var \left[Z\right]$.

Suppose the pair $\left\{X , Y\right\}$ is independent, with $X \sim$ gamma (3,0.1) and

$Y \sim$ Poisson (13). Let $Z = 2 X - 5 Y$. Determine $E \left[Z\right]$ and $Var \left[Z\right]$.

The pair $\left\{X , Y\right\}$ is jointly distributed with the following parameters:

Determine $Var \left[3 X - 2 Y\right]$.

For the Beta $\left(r , s\right)$ distribution,

1. Determine $E \left[X^{n}\right]$, where n is a positive integer.
2. Use the result of part (a) to determine $E \left[X\right]$ and $Var \left[X\right]$.

The pair $\left\{X , Y\right\}$ has joint distribution. Suppose

Determine $Var \left[15 X - 2 Y\right]$.

The pair $\left\{X , Y\right\}$ has joint distribution. Suppose

Determine $Var \left[3 X + 2 Y\right]$.

The pair $\left\{X , Y\right\}$ is independent, with

Let $Z = 2 X^{2} + X Y^{2} - 3 Y + 4.$.
Determine $E \left[\right. Z ].$

(See Exercise 9 from "Problems on Mathematical Expectation"). Random variable X has density function

$E \left[X\right] = 11 / 10$. Determine $Var \left[X\right]$.

(See Exercise 10 from "Problems On Random Vectors and Joint Distributions", and Exercise 20 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = 1$ for $0 \leq t \leq 1$, $0 \leq u \leq 2 \left(1 - t\right)$.

(See Exercise 13 from "Problems On Random Vectors and Joint Distributions", and Exercise 23 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = \frac{1}{8} \left(t + u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 2$.

(See Exercise 15 from "Problems On Random Vectors and Joint Distributions", and Exercise 25 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 1 + t$.

(See Exercise 16 from "Problems On Random Vectors and Joint Distributions", and Exercise 26 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = 12 t^{2} u$ on the parallelogram with vertices

(See Exercise 17 from "Problems On Random Vectors and Joint Distributions", and Exercise 27 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 , 2 - t\right\}$.

(See Exercise 18 from "Problems On Random Vectors and Joint Distributions", and Exercise 28 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \max \left\{2 - t , t\right\}$.

(See Exercise 21 from "Problems On Random Vectors and Joint Distributions", and Exercise 31 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = \frac{2}{13} \left(t + 2 u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{2 t , 3 - t\right\}$.

(See Exercise 22 from "Problems On Random Vectors and Joint Distributions", and Exercise 32 from "Problems on Mathematical Expectation"). $f_{X Y} \left(t , u\right) = I_{\left[0 , 1\right]} \left(t\right) \frac{3}{8} \left(t^{2} + 2 u\right) + I_{\left(1 , 2\right]} \left(t\right) \frac{9}{14} t^{2} u^{2}$, for $0 \leq u \leq 1$.

$f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 1 + t$ (see Exercise 25 and Exercise 37 from "Problems on Mathematical Expectation").

Determine $Var \left[Z\right]$ and $Cov \left[X , Z\right]$. Check with discrete approximation.

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 , 2 - t\right\}$ (see Exercise 27 and Exercise 38 from "Problems on Mathematical Expectation").

Determine $Var \left[Z\right]$ and $Cov \left[X , Z\right]$. Check with discrete approximation.

$f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \max \left\{2 - t , t\right\}$ (see Exercise 28 and Exercise 39 from "Problems on Mathematical Expectation").

Determine $Var \left[Z\right]$ and $Cov \left[Z\right]$. Check with discrete approximation.

$f_{X Y} \left(t , u\right) = \frac{12}{179} \left(3 t^{2} + u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{2 , 3 - t\right\}$ (see Exercise 29 and Exercise 40 from "Problems on Mathematical Expectation").

Determine $Var \left[Z\right]$ and $Cov \left[X , Z\right]$. Check with discrete approximation.

$f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 + t , 2\right\}$ (see Exercise 30 and Exercise 41 from "Problems on Mathematical Expectation").

Determine $Var \left[Z\right]$ and $Cov \left[X , Z\right]$. Check with discrete approximation.

(See the exercise, and Exercises 9 and 10 from "Problems on Functions of Random Variables"). For the pair $\left\{X , Y\right\}$ in the exercise, let

Determine the joint distribution for the pair $\left\{Z , W\right\}$ and determine the regression line of W on Z.