12.3 Problems on variance, covariance, linear regression  (Page 2/5)

The class $\{A,B,C,D,E,F\}$ 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 $E\left[X\right]$ , $\mathrm{Var}\left[X\right]$ , $E\left[Y\right]$ , and $\mathrm{Var}\left[Y\right]$ . Note that it is not necessary to obtain the distributions for X or Y .
2. Let $Z=3Y-2X$ .
   Determine $E\left[Z\right]$ , and $\mathrm{Var}\left[Z\right]$ .

Consider $X=-3.3I_A-1.7I_B+2.3I_C+7.6I_D-3.4$ . The class $\{A,B,C,D\}$ has minterm

probabilities (data are in m-file npr12_10.m )

1. Calculate $E\left[X\right]$ and $\mathrm{Var}\left[X\right]$ .
2. Let $W=2X^2-3X+2$ .
   Calculate $E\left[W\right]$ and $\mathrm{Var}\left[W\right]$ .

Consider a second random variable $Y=10I_E+17I_F+20I_G-10$ in addition to that in [link] . The class $\{E,F,G\}$ has minterm probabilities (in mfile npr12_10.m )

The pair $\{X,Y\}$ is independent.

1. Calculate $E\left[Y\right]$ and $\mathrm{Var}\left[Y\right]$ .
2. Let $Z=X^2+2XY-Y$ .
   Calculate $E\left[Z\right]$ and $\mathrm{Var}\left[Z\right]$ .

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

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

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

Determine $\mathrm{Var}[3X-2Y]$ .

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

Let

1. Use properties of expectation and variance to obtain $E\left[X\right]$ , $\mathrm{Var}\left[X\right]$ , $E\left[Y\right]$ , and $\mathrm{Var}\left[Y\right]$ .
2. Determine $E\left[Z\right]$ , and $\mathrm{Var}\left[Z\right]$ .
3. Use appropriate m-programs to obtain $E\left[X\right]$ , $\mathrm{Var}\left[X\right]$ , $E\left[Y\right]$ , $\mathrm{Var}\left[Y\right]$ , $E\left[Z\right]$ , and $\mathrm{Var}\left[Z\right]$ . Compare with results of parts (a) and (b).