Statistics - Probability Projects

3.37 : The distribution function of the random variable X is given

Problem 3.37 : The distribution function of the random variable $X$ is given by

\[F(x)=\left\{ \begin{aligned} & 1-(1+x){{e}^{-x}}\,\text{ for }x>0 \\ & 0\text{ for }x\le 0 \\

\end{aligned} \right.\]

Find $\Pr (X\le 2)$, $\Pr (1<X<3)$ and $P(X>4)$.

Problem 4.6: Find the expected value of the discrete random variable $X$, having the probability distribution

\[f(x)=\frac{|x-2|}{7}\] , for \[x=-1,0,1,3\]

Problem 4.7: Find the expected value of the random variable Y whose probability density is given by:

\[f(y)=\left\{ \begin{aligned}

& \frac{1}{8}(y+1)\text{ for }2<y<4 \\ & 0\text{ elsewhere} \\

\end{aligned} \right.\]

Problem 4.10 :

If the probability density of $X$ is given by
\[f(x)=\left\{ \begin{aligned}
& \frac{1}{x\ln 3}\text{ for 1}<x<3 \\
& \text{ }0\text{ elsewhere} \\
\end{aligned} \right.\]
find $E(X)$, $E({{X}^{2}})$ and $E({{X}^{3}})$.
Use the results of part (a) to determine $E({{X}^{3}}+2{{X}^{2}}-3X+1)$.

Problem 4.23: If the random variable X has mean $\mu $ and standard deviation $\sigma $, show that the random variable Z whose values are related to those of X by means of the equation

\[Z=\frac{X-\mu }{\sigma }\]

has $E(Z)=0$, $\operatorname{var}(Z)=1$.

Problem 4.33: Find the moment-generating function of the discrete random variable with probability distribution:

\[f(x)=2{{\left( \frac{1}{3} \right)}^{x}}\] for \[x=1,2,3,4.....\]

and use it to find $\mu _{1}^{'}$ and $\mu _{2}^{'}$

Problem 4.37: Show that if a random variable has the probability density

\[f(x)=\frac{1}{2}{{e}^{-|x|}}\] for \[-\infty <x<\infty \]

its moment generating function is given by:

\[{{M}_{X}}(t)=\frac{1}{1-{{t}^{2}}}\]

Problem 4.41: If $X$ and $Y$ have the joint distribution $f(x,y)=\frac{1}{4}$ for $x=-3$ and $y=-5$, $x=-1$ and $y=-1$, $x=1$ and $y=1$, and $x=3$ and $y=5$, find $\operatorname{cov}(X,Y)$.

Problem 4.48: If ${{X}_{1}},{{X}_{2}}$, and ${{X}_{3}}$ are independent and have the means 4, 9 and 3, and the variances 3, 7, and 5, find the mean and variance of

$Y=2{{X}_{1}}-3{{X}_{2}}+4{{X}_{3}}$
$Z={{X}_{1}}+2{{X}_{2}}-{{X}_{3}}$

Problem 4.52: Express $\operatorname{var}(X+Y)$, $\operatorname{var}(X-Y)$ and $\operatorname{cov}(X+Y,X-Y)$ in terms of the variances and covariance of $X$ and Y.

Problem 4.59:

Show that the conditional distribution function if the continuous random variable X , given $a<X\le b$ is given by
\[F(x|a<X\le b)=\left\{ \begin{aligned}
& 0\text{ for }x\le a \\
& \frac{F(x)-F(a)}{F(b)-F(a)}\text{ for }a<x\le b \\
& 1\text{ for }x>b \\
\end{aligned} \right.\]
Differentiate the above result with respect to x to find the conditional density of $X$ given $a<X\le b$, and show that

\[E(u(X)|a<X\le b)=\frac{\int\limits_{a}^{b}{u(x)f(x)dx}}{\int\limits_{a}^{b}{f(x)dx}}\]

Price: $15.89

Solution: The downloadable solution consists of 11 pages, 489 words.
Deliverable: Word Document