Maharashtra Board Solutions for Chapter: Probability Distributions, Exercise 2: EXERCISE 7.2

Verify the following is p.d.f. of r.v. $X$ :
$f\left(x\right)=x$, for $0\le x\le 1$ and $=2-x$ for $1<x<2$

Verify the following is p.d.f. of r.v. $X$ :
$f\left(x\right)=2$, for $0\le x\le 1$

Suppose that $X$ is waiting time in minutes for a bus and its p.d.f. is given by $f\left(x\right)=\frac{1}{5}$, for $0\le x\le 5$ and $=0$ otherwise.
Find the probability that
(i) waiting time is between $1$ and $3$
(ii) waiting time is more than $4$minutes.

Suppose error involved in making a certain measurement is continuous r.v. $X$ with p.d.f.

$f\left(x\right)=k\left(4-x^2\right)$, for $-2\le x\le 2$ and $=0$0$ otherwise.
Compute : (i)$P(x>0)$
(ii) $P(-1<x<1)$
(iii) $P(-0.5<x$ or $x>0.5)$

The following is the p.d.f. of continuous r.v. $f\left(x\right)=\frac{x}{8}$, for $0<x<4$ and $=0$ otherwise 
(i) Find expression for c.d.f. of $X$

The following is the $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ of continuous $\mathrm{r}.\mathrm{v}.$ $f\left(x\right)=\frac{x}{8}$, for $0<x<4$ and $=0$ otherwise.

Find $F\left(x\right)$ at $x=0.5,1.7$ and $5$

Given the $\mathrm{p}.\mathrm{d}.\mathrm{f}.$ of a continuous $\mathrm{r}.\mathrm{v}.$ $X,f\left(x\right)=\frac{x^2}{3}$, for $-1<x<2$ and $=0$ otherwise Determine $\mathrm{c}.\mathrm{d}.\mathrm{f}.$ of $X$ hence find $P\left(x<1\right),P\left(x<-2\right),P(\mathrm{X}>0),P(1<\mathrm{x}<2)$

If a $\mathrm{r}.\mathrm{v}.$ $X$  has $\mathrm{p}.\mathrm{d}.\mathrm{f}.$, $f\left(x\right)=\frac{c}{x}$, for $1<x<3, c>0$, Find $c,E\left(X\right)$ and Var $\left(X\right)$.