(C) The cumulative distribution function $F(x)$ is defined as $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$. For $-1 < x < 2$,we have: $F(x) = \in

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(C) The cumulative distribution function $F(x)$ is defined as $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$. For $-1 $F(x) = \int_{-1}^{x} \frac{t^3}{3} dt$ $F(x) = \frac{1}{3} \left[ \frac{t^4}{4} \right]_{-1}^{x}$ $F(x) = \frac{1}{3} \left( \frac{x^4}{4} - \frac{(-1)^4}{4} \right)$ $F(x) = \frac{1}{3} \left( \frac{x^4}{4} - \frac{1}{4} \right)$ $F(x) = \frac{1}{12} (x^4 - 1)$

Class 12 Mathematics Probability Probability distribution

Medium

If the probability density function of a continuous random variable $X$ is $f(x) = \frac{x^3}{3}$ for $-1 < x < 2$,and $f(x) = 0$ otherwise,then the cumulative distribution function $F(x)$ for $-1 < x < 2$ is:

MHT CET 2020 All MHT CET

A
$\frac{1}{14}(x^4 - 1)$
B
$\frac{1}{10}(x^4 - 1)$
C
$\frac{1}{12}(x^4 - 1)$
D
$\frac{1}{16}(x^4 - 1)$

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If $X$ is a random variable with probability distribution $P(X=k) = \frac{(k+1)c}{2^k}$ for $k = 0, 1, 2, \ldots$,then $P(X \geq 3) = $

MediumAP EAMCET 2023

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Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0) = \frac{1}{2}$ and $P(X=j) = \frac{1}{3^j}$ for $j = 1, 2, 3, \ldots, \infty$. Then the mean of the distribution and $P(X \text{ is positive and even})$ respectively are:

MediumJEE MAIN 2021

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Let $X$ be a random variable representing the number of heads obtained when three fair coins are tossed simultaneously. Find the value of $P(X = 2)$.

EasyAP EAMCET 2020

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If a random variable $x$ has the probability distribution as follows:
$x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$P(x)$ $0$ $2k$ $k$ $3k$ $2k^2$ $2k$ $k^2+k$ $7k^2$

Then $P(3 < x \leq 6)$ is equal to:

DifficultJEE MAIN 2026

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Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X = x)$ $0.1$ $0.15$ $0.3$ $0.25$ $k$ $k$

Find the variance of $X$.

MediumTS EAMCET 2020

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

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$0.1$	$0.15$	$0.3$	$0.25$	$k$	$k$

If the probability density function of a continuous random variable $X$ is $f(x) = \frac{x^3}{3}$ for $-1 < x < 2$,and $f(x) = 0$ otherwise,then the cumulative distribution function $F(x)$ for $-1 < x < 2$ is:

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If $X$ is a random variable with probability distribution $P(X=k) = \frac{(k+1)c}{2^k}$ for $k = 0, 1, 2, \ldots$,then $P(X \geq 3) = $

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If a random variable $x$ has the probability distribution as follows:
$x$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
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Then $P(3 < x \leq 6)$ is equal to:

Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X = x)$ $0.1$ $0.15$ $0.3$ $0.25$ $k$ $k$

Find the variance of $X$.