(A) For a probability density function,the total area under the curve must be $1$. $\int_{0}^{4} f(x) dx = 1$ $\int_{0}^{4} \frac{k}{\sqrt{x}} dx = 1$

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(A) For a probability density function,the total area under the curve must be $1$. $\int_{0}^{4} f(x) dx = 1$ $\int_{0}^{4} \frac{k}{\sqrt{x}} dx = 1$ $k [2\sqrt{x}]_{0}^{4} = 1$ $k [2(2) - 0] = 1 \Rightarrow 4k = 1 \Rightarrow k = \frac{1}{4}$ Now,we calculate $P(1 $P(1 $= \frac{1}{4} [2\sqrt{x}]_{1}^{4}$ $= \frac{1}{2} [\sqrt{4} - \sqrt{1}]$ $= \frac{1}{2} [2 - 1] = \frac{1}{2}$

Class 12 Mathematics Probability Probability distribution

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The p.d.f. of a random variable $X$ is given by $f(x) = \frac{k}{\sqrt{x}}$ for $0 \leq x \leq 4$ and $f(x) = 0$ otherwise. Then $P(1 < X < 4) = $

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A
$\frac{1}{2}$
B
$\frac{1}{3}$
C
$\frac{1}{5}$
D
$\frac{3}{4}$

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Probability

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Probability distribution

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \dots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?
Outcome $\omega_1$ $\omega_2$ $\omega_3$ $\omega_4$ $\omega_5$ $\omega_6$
$(e)$ $0.1$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$

Easy

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If the probability distribution of a random variable $X$ is as follows,then the mean of $X$ is:
$X = x_i$ $-1$ $0$ $1$ $2$
$P(X = x_i)$ $k^3$ $2k^3 + k$ $4k - 10k^2$ $4k - 1$

$X = x_i$	$-1$	$0$	$1$	$2$
$P(X = x_i)$	$k^3$	$2k^3 + k$	$4k - 10k^2$	$4k - 1$

MediumAP EAMCET 2025

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An urn contains $5$ red and $2$ black balls. Two balls are randomly drawn. Let $X$ represent the number of black balls. What are the possible values of $X?$ Is $X$ a random variable?

Easy

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$A$ fair coin is tossed $4$ times. If $X$ is a random variable which indicates the number of heads,then $P[X < 3] = $

DifficultMHT CET 2021

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$A$ 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:

DifficultAP EAMCET 2003

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Outcome	$\omega_1$	$\omega_2$	$\omega_3$	$\omega_4$	$\omega_5$	$\omega_6$
$(e)$	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$

The p.d.f. of a random variable $X$ is given by $f(x) = \frac{k}{\sqrt{x}}$ for $0 \leq x \leq 4$ and $f(x) = 0$ otherwise. Then $P(1 < X < 4) = $

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$A$ 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:

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \dots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?
Outcome $\omega_1$ $\omega_2$ $\omega_3$ $\omega_4$ $\omega_5$ $\omega_6$
$(e)$ $0.1$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$

If the probability distribution of a random variable $X$ is as follows,then the mean of $X$ is:
$X = x_i$ $-1$ $0$ $1$ $2$
$P(X = x_i)$ $k^3$ $2k^3 + k$ $4k - 10k^2$ $4k - 1$