Maharashtra Board Solutions for Chapter: Probability Distributions, Exercise 3: MISCELLANEOUS EXERCISE 7

The following is the c.d.f of R.V $X$

$\begin{array}{|ccccccccc|}\hline X& -3& -2& -1& 0& 1& 2& 3& 4\\ F\left(X\right)& 0.1& 0.3& 0.5& 0.65& 0.75& 0.85& 0.9& 1\\ \hline\end{array}$

Find $P\left(-1\le X\le 2\right)$

The following is the c.d.f of R.V $X$

$\begin{array}{|ccccccccc|}\hline X& -3& -2& -1& 0& 1& 2& 3& 4\\ F\left(X\right)& 0.1& 0.3& 0.5& 0.65& 0.75& 0.85& 0.9& 1\\ \hline\end{array}$

Find $P(X\le 3\mid X>0)$

A player tosses two coins he wins $₹ 10$ if $2$ heads appears, $₹ 5$ if $1$ head appears and $₹ 2$ if no head appears. Find the expected winning amount and variance of winning amount.

Let the $\mathrm{p}.\mathrm{m}.\mathrm{f}.$ of $\mathrm{r}.\mathrm{v}.$ $X$ be $P\left(x\right)=\frac{3-x}{10}$, for $x=-1,0,1,2$ and $=0$, otherwise Calculate $E\left(x\right)$ and $Var\left(X\right)$.

Suppose error involved in making a certain measurement is continuous $\mathrm{r}.\mathrm{v}. x$  with $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ $f\left(x\right)=k\left(4-x^2\right)$, for $-2\le x\le 2$ and $=0$ otherwise.
Compute $P\left(X>0\right)$

Suppose error involved in making a certain measurement is continuous $\mathrm{r}.\mathrm{v}. x$  with $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ $f\left(x\right)=k\left(4-x^2\right)$, for $-2\le x\le 2$ and $=0$ otherwise.
Compute $P\left(-1<x<1\right)$

Suppose error involved in making a certain measurement is continuous $\mathrm{r}.\mathrm{v}. x$  with $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ $f\left(x\right)=k\left(4-x^2\right)$, for $-2\le x\le 2$ and $=0$ otherwise.
Compute $P\left(X<0.5 \mathrm{or} X>0.5\right)$

The $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ of $\mathrm{r}.\mathrm{v}.$ of $X$ is given by $f\left(x\right)=\frac{k}{\sqrt{x}}$, for $0<x<4$ and $=0$, otherwise. Determine $k$. Determine $\mathrm{c}.\mathrm{d}.\mathrm{f}.$ of $X$ and hence $P\left(X\le 2\right)$ and $P\left(X\le 1\right)$.