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 $y=m x+c$

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

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

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

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

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

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

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

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

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

$\therefore 24 \alpha=42$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Solution of the equation
$sec^2x\ tany\,dx + sec^2y\ tanx\,dy = 0,y \left( {\frac{\pi }{4}} \right) = \frac{\pi}{3}$ is :-

A six faced fair dice is thrown until $2$ comes, then the probability that $2$ comes in even number of trials is (dice having six faces numbered $1, 2, 3, 4, 5$ and $6$)

The curve $y(x)=a x^{3}+b x^{2}+c x+5$ touches the $x$-axis at the point $P (-2,0)$ and cuts the $y$-axis at the point $Q$, where $y ^{\prime}$ is equal to $3$ . Then the local maximum value of $y ( x )$ is.

If $\sqrt {3{x^2} - 7x - 30} + \sqrt {2{x^2} - 7x - 5} = x + 5$,then $x$ is equal to

The correct evaluation of $\int_0^{\pi /2} {\left| {\,\sin \left( {x - \frac{\pi }{4}} \right)\,} \right|\,dx} $ is

The moment about the point $M( - 2,\,4,\, - 6)$ of the force represented in magnitude and position by $\overrightarrow {AB} $ where the points $A$ and $ B$ have the co-ordinates $(1,\,2,\, - 3)$ and $(3,\, - 4,\,2)$ respectively, is

The number of triplets $(x, y, z)$. where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is

The value of $\mathrm{k} \in \mathbb{N}$ for which the integral

$I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N} \text {, satisfies } 147 I_{20}=148 I_{21}$ is :

The function $'f'$ is defined by $f(x) = \left\{ {\begin{array}{*{20}{c}}{ 2x - 1,}&{if \,\,\, x > 2}\\{k,}&{ if \,\,\,x=2}\\{{x^2} - 1,}&{if \,\,\, x < 2 }\end{array}} \right.$ is continuous, then the value of $k$ is equal to

$\sqrt {\frac{{1 - \sin A}}{{1 + \sin A}}} = $