Chapter 2

Let ${\sigma }_1^2={\sigma }_2^2={\sigma }^2$ be the common variance of $X_1$ and $X_2$ and let $\rho$ be the correlation coefficient of $X_1$ and $X_2$. Show that $$P[|(X_1-{\mu }_1)+(X_2-{\mu }_2)|\ge k\sigma ]\le \frac{2(1+\rho )}{k^2}$$

Let $f(x)$ and $F(x)$ denote, respectively, the pdf and the cdf of the random variable $X$. The conditional pdf of $X$, given $X>x_0,x_0$ a fixed number, is defined by \(f\left(x \mid X>x_{0}\right)=f(x) /\left[1-F\left(x_{0}\right)\right], x_{0}x_{0}\right)\) is a pdf. (b) Let \(f(x)=e^{-x}, 02 \mid X>1)\).

Let $X_1,X_2$ be two random variables with joint $\mathrm{pmf}p(x_1,x_2)=(x_1+x_2)/12$, for $x_1=1,2,x_2=1,2$, zero elsewhere. Compute $E\left(X_1\right),E\left(X_1^2\right),E\left(X_2\right),E\left(X_2^2\right)$, and $E\left(X_1{X}_2\right).$ Is $E\left(X_1{X}_2\right)=E\left(X_1\right)E\left(X_2\right)?$ Find $E(2X_1-6X_2^2+7X_1{X}_2)$

Cast a fair die and let $X=0$ if 1,2, or 3 spots appear, let $X=1$ if 4 or 5 spots appear, and let $X=2$ if 6 spots appear. Do this two independent times, obtaining $X_1$ and $X_2.$ Calculate $P(|X_1-X_2|=1)$.

Let $X_1,X_2$ be two random variables with joint pdf $f(x_1,x_2)=4x_1{x}_2$ \(0

Let $X_1,X_2$ be two random variables with joint $\mathrm{pmf}p(x_1,x_2)=(1/2{)}^{x_1+x_2}$, for $1\le x_i<\mathrm{\infty },i=1,2$, where $x_1$ and $x_2$ are integers, zero elsewhere. Determine the joint mgf of $X_1,X_2$. Show that $M(t_1,t_2)=M(t_1,0)M(0,t_2)$.

Let $X_1,X_2$ be two random variables with joint pdf $\text{Missing left or extra right}$, for \(0

Let $X$ and $Y$ have the joint pdf \(f(x, y)=6(1-x-y), x+y<1,0