A random variable $X$ has the following probability distribution:
$X$ 0 1 2 3 4 5
$P (X = x)$ $\frac{1}{4}$ $2 a$ $3 a$ $4 a$ $5 a$ $\frac{1}{4}$
Then $P (1 \leq X \leq 4)$ is:

\frac{10}{21}

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\frac{2}{7}

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\frac{1}{14}

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\frac{1}{2}

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Solution

The correct option is C $\frac{1}{2}$
We know that $\sum P (X = x) = 1$

$\Rightarrow \frac{1}{4} + 2 a + 3 a + 4 a + 5 a + \frac{1}{4} = 1$

$\Rightarrow \frac{1}{2} + 14 a = 1$

$\Rightarrow 14 a = 1 - \frac{1}{2}$

$\Rightarrow 14 a = \frac{1}{2}$

$\Rightarrow a = \frac{1}{28}$

Now we have to find $P (1 \leq x \leq 4)$

$P (1 \leq x \leq 4) = P (x = 1) + P (x = 2) + P (x = 3) + P (x = 4)$

$= 2 a + 3 a + 4 a + 5 a$

$= 14 a$

$= 14 (\frac{1}{28}) [∵ a = \frac{1}{28}]$

$= \frac{1}{2}$

$∴ P (1 \leq x \leq 4) = \frac{1}{2}$

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$X$	0	1	2	3	4	5
$P (X = x)$	$\frac{1}{4}$	$2 a$	$3 a$	$4 a$	$5 a$	$\frac{1}{4}$