Let X be a random variable, and let P(X=x) denote the probability… - Vidyadip

MCQ

Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x, P(X=x)), x=0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P(X=x)=0$ for all $x \in R$ $\{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is. . . . .

A
$20$
B
$30$
C
$40$
✓
$42$

✓

Answer

Correct option: D.

$42$

d
Let equation of line is $y=m x+c$

$x$	$0$	$1$	$2$	$3$	$4$	$R -\{0,1,2,3,4\}$
$P ( x )$	$C$	$m + c$	$2 m + c$	$3 m + c$	$4 m+c$	$0$

$\sum_{ x =0}^4( mx + c )=1 \Rightarrow 10 m +5 c =1 \Rightarrow 2 m + c =\frac{1}{5}$ $. . . (1)$

$\text { mean }=\sum x _{ i } P _{ i }=\sum_{ i =0}^4\left( mx _{ i }+ c \right) x _{ i }=30 m +10 c =\frac{5}{2}$

$\therefore 3 m + c =\frac{1}{4} \ldots(2)$

$\text { from (1) and (2) m= } \frac{1}{20}, c =\frac{1}{10}$

$\sum P _{ i } x _{ i }^2=\sum_{ i =0}^4\left( mx _{ i }+ c \right) x _1^2$

$=\sum_{ i =0}^4\left( mx _{ i }^3+ cx _{ i }^2\right) \Rightarrow 100 m +30 c (\text { Now putting } m \text { and } c )$

$\Rightarrow \Sigma P _{ i }^2=5+3=8$

$\text { Variance }=\Sigma P _{ i } x _{ i }^2-\left(\Sigma P _{ i } x _{ i }\right)^2=8-\left(\frac{5}{2}\right)^2=\frac{7}{4}$

$\therefore 24 \alpha=42$

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