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 $x > 0$. $F(x) = \int_{0}^{x} \frac{2/\pi}{t^2+1} dt = \frac{2}{\pi} [	an^{-1} t]_{0}^{x} = \frac{2}{\pi} 	an^{-1} x$.

$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X=x)$	$k$	$2k$	$3k$	$4k$	$4k$	$3k$	$2k$	$k$	$k$

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$0$	$k$	$2k$	$3k$	$3k^2$	$k^2$	$2k^2$	$7k^2+k$

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X)$	$k$	$3k$	$5k$	$7k$	$9k$	$11k$	$13k$

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

$A$ random variable $X$ has the following probability distribution:
$x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$P(X=x)$ $k$ $2k$ $3k$ $4k$ $4k$ $3k$ $2k$ $k$ $k$

Then $P(3 < X \leq 6) = $

$A$ player tosses two coins. He wins $Rs. 1$ if $1$ head appears,$Rs. 2$ if $2$ heads appear. But he loses $Rs. 3$ if no head appears. The mean of the prize money is

$A$ random variable $X$ has the following probability distribution:

$X$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$

$P(X)$ $0$ $k$ $2k$ $3k$ $3k^2$ $k^2$ $2k^2$ $7k^2+k$

Determine $P(X < 3)$. (in $/10$)

$A$ random variate $X$ takes the values $0, 1, 2, 3$ and its mean is $1.3$. If $P(X=3) = 2 P(X=1)$ and $P(X=2) = 0.3$,then $P(X=0)$ is equal to:

$A$ random variable $X$ has the following probability distribution:
$X$ $0$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X)$ $k$ $3k$ $5k$ $7k$ $9k$ $11k$ $13k$

Then find $P(X \ge 2)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module