(C) Since $f(x)$ is the probability density function (p.d.f.) of $X$,the total area under the curve must be $1$. $\int_{-\infty}^{\infty} f(x) dx = 1$

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(C) Since $f(x)$ is the probability density function (p.d.f.) of $X$,the total area under the curve must be $1$. $\int_{-\infty}^{\infty} f(x) dx = 1$ $\int_0^1 ax dx + \int_1^2 a dx + \int_2^3 (3a - ax) dx = 1$ $a \left[ \frac{x^2}{2} \right]_0^1 + a [x]_1^2 + \left[ 3ax - \frac{ax^2}{2} \right]_2^3 = 1$ $a(\frac{1}{2}) + a(1) + [(9a - \frac{9a}{2}) - (6a - \frac{4a}{2})] = 1$ $\frac{a}{2} + a + [\frac{9a}{2} - 4a] = 1$ $\frac{a}{2} + a + \frac{a}{2} = 1$ $2a = 1$ $a = \frac{1}{2}$

Class 12 Mathematics Probability Probability distribution

Medium

If a continuous random variable $X$ has probability density function $f(x)$ given by $f(x) = \begin{cases} ax, & 0 \le x < 1 \\ a, & 1 \le x < 2 \\ 3a - ax, & 2 \le x \le 3 \\ 0, & \text{otherwise} \end{cases}$,then $a$ has the value:

MHT CET 2023 All MHT CET

A
$\frac{1}{5}$
B
$\frac{1}{3}$
C
$\frac{1}{2}$
D
$1$

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Consider the probability distribution
$\begin{array}{|r|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 \\ \hline P(X=x) & K & 2K & K^2 & 2K & 5K^2 \\ \hline \end{array}$
Then the value of $P(X > 2)$ is

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For the following cumulative distribution function $F(x)$ of a random variable $X$,find $P(3 < X \leq 5)$.
$x$ $1$ $2$ $3$ $4$ $5$ $6$
$F(x)$ $0.2$ $0.37$ $0.48$ $0.62$ $0.85$ $1$

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$F(x)$	$0.2$	$0.37$	$0.48$	$0.62$	$0.85$	$1$

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If $X$ is a Poisson variable representing the number of successes in $50$ trials such that $2 P(X=1) = 5 P(X=5) + 2 P(X=3)$,then the probability of getting success in one trial is

MediumTS EAMCET 2019

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The probability distribution of $X$ is
$\begin{array}{|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.3 & k & 2k & 2k \\ \hline \end{array}$
Find the value of $k$.

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$A$ 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 $P(0 < X < 3)$. (in $/10$)

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

If a continuous random variable $X$ has probability density function $f(x)$ given by $f(x) = \begin{cases} ax, & 0 \le x < 1 \\ a, & 1 \le x < 2 \\ 3a - ax, & 2 \le x \le 3 \\ 0, & \text{otherwise} \end{cases}$,then $a$ has the value:

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Consider the probability distribution$\begin{array}{|r|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 \\ \hline P(X=x) & K & 2K & K^2 & 2K & 5K^2 \\ \hline \end{array}$Then the value of $P(X > 2)$ is

For the following cumulative distribution function $F(x)$ of a random variable $X$,find $P(3 < X \leq 5)$.$x$$1$$2$$3$$4$$5$$6$$F(x)$$0.2$$0.37$$0.48$$0.62$$0.85$$1$

If $X$ is a Poisson variable representing the number of successes in $50$ trials such that $2 P(X=1) = 5 P(X=5) + 2 P(X=3)$,then the probability of getting success in one trial is

The probability distribution of $X$ is$\begin{array}{|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.3 & k & 2k & 2k \\ \hline \end{array}$Find the value of $k$.

$A$ 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 $P(0 < X < 3)$. (in $/10$)

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Consider the probability distribution
$\begin{array}{|r|c|c|c|c|c|} \hline X=x & 1 & 2 & 3 & 4 & 5 \\ \hline P(X=x) & K & 2K & K^2 & 2K & 5K^2 \\ \hline \end{array}$
Then the value of $P(X > 2)$ is

For the following cumulative distribution function $F(x)$ of a random variable $X$,find $P(3 < X \leq 5)$.
$x$ $1$ $2$ $3$ $4$ $5$ $6$
$F(x)$ $0.2$ $0.37$ $0.48$ $0.62$ $0.85$ $1$

The probability distribution of $X$ is
$\begin{array}{|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.3 & k & 2k & 2k \\ \hline \end{array}$
Find the value of $k$.

$A$ 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 $P(0 < X < 3)$. (in $/10$)