If a random variable X has the p.d.f. f(x) = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Step $1$: Find the value of $k$ using the property $\int_{-\infty}^{\infty} f(x) dx = 1$. Since $f(x) = 0$ for $x \leq 0$,we have $\int_{0}^{\inft

Vedclass · Accepted Answer

$\frac{2}{\pi} 	an^{-1} x$

Vedclass · Answer

(C) Step $1$: Find the value of $k$ using the property $\int_{-\infty}^{\infty} f(x) dx = 1$. Since $f(x) = 0$ for $x \leq 0$,we have $\int_{0}^{\infty} \frac{k}{x^2+1} dx = 1$. Step $2$: Evaluate the integral: $k [	an^{-1} x]_{0}^{\infty} = 1$. $k (\frac{\pi}{2} - 0) = 1 \implies k = \frac{2}{\pi}$. Step $3$: The c.d.f. $F(x)$ is defined as $P(X \leq x) = \int_{0}^{x} f(t) dt$ for $x > 0$. $F(x) = \int_{0}^{x} \frac{2/\pi}{t^2+1} dt = \frac{2}{\pi} [	an^{-1} t]_{0}^{x} = \frac{2}{\pi} 	an^{-1} x$.

$X=x$	$-1$	$0$	$1$	$2$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{3}$

If a random variable $X$ has the p.d.f. $f(x) = \begin{cases} \frac{k}{x^2+1} & , \text{if } 0 < x < \infty \\ 0 & , \text{otherwise} \end{cases}$,then the c.d.f. of $X$ is:

Similar Questions

Let $X$ be a random variable with probability distribution function,$P(X=x)=K\left(\frac{2}{5}\right)^x, x=1, 2, 3, \ldots$. Then,the value of $K$ is

The probability distribution of a discrete random variable $X$ is given below. If $E(X^2) = \Sigma x^2 P(X=x)$,then $6 E(X^2) - \operatorname{Var}(X) =$
$X=x$ $-1$ $0$ $1$ $2$
$P(X=x)$ $\frac{1}{3}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{3}$

From a bag containing $4$ white and $5$ red balls,if $3$ balls are drawn at random,then the mean of the number of red balls among the balls drawn is:

The probability that an individual suffers a bad reaction from an injection is $0.001$. The probability that out of $2000$ individuals exactly three will suffer a bad reaction is:

The range of a random variable $X$ is $\{1, 2, 3, \ldots\}$ and $P(X=x) = \frac{c^x}{x!}$ for $x = 1, 2, 3, \ldots$. Then the value of $c$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module