(A) Given the p.d.f. $f(x) = \frac{1}{2}$ for $0 < x < 2$. First,we calculate $a = P(X < \frac{1}{2})$: $a = \int_{0}^{1/2} \frac{1}{2} dx = \frac{1}{

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(A) Given the p.d.f. $f(x) = \frac{1}{2}$ for $0 First,we calculate $a = P(X $a = \int_{0}^{1/2} \frac{1}{2} dx = \frac{1}{2} [x]_{0}^{1/2} = \frac{1}{2} (\frac{1}{2} - 0) = \frac{1}{4}$. Next,we calculate $b = P(X > \frac{3}{2})$: $b = \int_{3/2}^{2} \frac{1}{2} dx = \frac{1}{2} [x]_{3/2}^{2} = \frac{1}{2} (2 - \frac{3}{2}) = \frac{1}{2} (\frac{1}{2}) = \frac{1}{4}$. Comparing $a$ and $b$,we have $a = \frac{1}{4}$ and $b = \frac{1}{4}$. Therefore,$a - b = \frac{1}{4} - \frac{1}{4} = 0$.

Class 12 Mathematics Probability Probability distribution

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The p.d.f. of a continuous random variable $X$ is given by $f(x) = \frac{1}{2}$ if $0 < x < 2$ and $f(x) = 0$ otherwise. If $a = P(X < \frac{1}{2})$ and $b = P(X > \frac{3}{2})$,then the relation between $a$ and $b$ is:

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A
$a - b = 0$
B
$2a - b = 0$
C
$3a - b = 0$
D
$a - 2b = 0$

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$A$ random variable $X$ takes values $-1, 0, 1, 2$ with probabilities $\frac{1+3p}{4}, \frac{1-p}{4}, \frac{1+2p}{4}, \frac{1-4p}{4}$ respectively,where $p$ varies over $\mathbb{R}$. Then the minimum and maximum values of the mean of $X$ are respectively.

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If a random variable $X$ has the following probability distribution,then the mean of $X$ is
$X = x_i$ $1$ $2$ $3$ $5$
$P(X = x_i)$ $2k^2$ $k$ $k$ $k^2$

$X = x_i$	$1$	$2$	$3$	$5$
$P(X = x_i)$	$2k^2$	$k$	$k$	$k^2$

EasyTS EAMCET 2024

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

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

MediumMHT CET 2017

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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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$A$ random variable $X$ has the following probability distribution:
$X$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
$P(X=x)$ $0.15$ $0.23$ $0.12$ $0.10$ $0.20$ $0.08$ $0.07$ $0.05$

For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 4\}$,find $P(E \cup F)$.

$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X=x)$	$0.15$	$0.23$	$0.12$	$0.10$	$0.20$	$0.08$	$0.07$	$0.05$

MediumMHT CET 2024

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

The p.d.f. of a continuous random variable $X$ is given by $f(x) = \frac{1}{2}$ if $0 < x < 2$ and $f(x) = 0$ otherwise. If $a = P(X < \frac{1}{2})$ and $b = P(X > \frac{3}{2})$,then the relation between $a$ and $b$ is:

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If a random variable $X$ has the following probability distribution,then the mean of $X$ is
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For the following cumulative distribution function $F(x)$ of a random variable $X$,find $P(3 < X \leq 5)$.
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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:
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$P(X=x)$ $0.15$ $0.23$ $0.12$ $0.10$ $0.20$ $0.08$ $0.07$ $0.05$

For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 4\}$,find $P(E \cup F)$.