Continuous Distributions

If $F\left(x\right)$ is the cumulative distributive function of a random variable $x$ whose range is from $-a to +a$, then $P(X<-a)=$

The value of $K$ for which the probability density function of a variate $X$ is given below.

Explain probability density function with an example.

A continuous random variable, $X$, has the following probability function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}k\left(x-2\right) 2⩽x⩽5\\ k\left(7-x\right) 5⩽x⩽7\\ 0 \mathrm{otherwise}\end{array}\right.$

Here, $k=\frac{2}{13}$.

Find the value of $x$ in surd, such that $P\left(X<x\right)=20\%$.

Find the mean and standard deviation in which $8\%$ items are under the $72$ and $99\%$ are over for the $53$.

A normal distribution has a mean measure $150$ and standard deviation as $10$. Find the value of middle $60\%$ of limits lies between which numbers.

A given set of $4$ coins are tossed $64$ times. The Number of occurrence of heads the table as given below

Then find a fitted Binomial distribution for the said data and find the expected frequencies for the given data.

Probability density function is always

Probability density function is associated with

A continuous random variable, $X$, has the following probability function:

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}k\left(x-2\right) 2⩽x⩽5\\ k\left(7-x\right) 5⩽x⩽7\\ 0 \mathrm{otherwise}\end{array}\right.$

Here, $k=\frac{2}{13}$.

Find the value of $x$ such that $P\left(X<x\right)=20\%$, giving your answer in the surd form:

A continuous random variable, $X$, has the following probability density function:

$f\left(x\right)=\left\{\begin{array}{l}\frac{2}{25}\left(5-x\right), 0⩽x⩽5\\ 0, \text{otherwise}\end{array}\right.$

Find $P\left(X⩽2\right)$.

The pdf of continuous random variable $x$ is given by  $f\left(x\right)=\left\{\begin{array}{l}\frac{x}{8},0<x<4\\ 0\text{ ,otherwise }\end{array}\right.$

Find, i) $P(x\le 2)$  ii) $P(2<x\le 3)$    iii) $P(x>3)$

Determine $k$ if $f\left(x\right)=\left\{\begin{array}{cl}k{e}^{-\theta x}& , 0\le x<\infty ,\theta >0\\ 0& ,\text{ otherwise }\end{array}\right.$  is the p.d.f. of the$r.v. X.$

Also find $P\left(X>\frac{1}{\theta }\right)$ and find $\mathrm{M}$ if $P(0<X<M)=\frac{1}{2}$

Suppose$r.v. \mathrm{X} =$ Waiting time in minutes for a bus by a passenger and its p.d.f. is given by

$f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{5},& 0\le x\le 5\\ 0& \text{ otherwise }\end{array}\right.$, Find the probability that waiting time is more than$4 \text{minutes}.$

Suppose random variable$\left\{X\right\}=$ Waiting time in minutes for a bus by a passenger and its probability density function is given by $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{5},& 0\le x\le 5\\ 0& \text{ otherwise }\end{array}\right.$. Find probability that waiting time is between$1$and $3$minutes, 

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ll}k\left(4-x^2\right),& -2\le x\le 2\\ 0,& \mathrm{otherwise}\end{array}\right.$

Compute $P(X<-0.5\text{ or }X>0.5)$.

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ll}k\left(4-x^2\right),& -2\le x\le 2\\ 0,& \mathrm{otherwise}\end{array}\right.$

Compute $P(-1<X<1)$   [Write your answer as lowest fraction form]

Suppose the error involved in making a certain measurement is a continuous $r. v. \mathrm{X}$ with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ccc}k\left(4-x^2\right)& ,& -2\le x\le 2\\ 0& ,& \text{ otherwise. }\end{array}\right.$

$P(X>0)=\frac{a}{b}$. Find $a+b$.

Let $X=$ time (in minutes) that elapses between the bell and the end of the lecture in case of a college professor. Suppose $X$ has p.d.f.

$f\left(x\right)=\left\{\begin{array}{lll}k{x}^2& ,& 0\le x\le 2\\ 0& ,& \mathrm{otherwise}\end{array}\right.$

Probability that lecture ends within $1$ minute of the bell ringing is $\frac{a}{b}$, then find the value of $b-a$.

Let $X=$ time (in minutes) that elapses between the bell and the end of the lecture in case of a college professor. Suppose $X$ has p.d.f.

$f\left(x\right)=\left\{\begin{array}{lll}k{x}^2& ,& 0\le x\le 2\\ 0& ,& \mathrm{otherwise}\end{array}\right.$. If $k=\frac{a}{b}$, then find the value of $a+b$.