Download App

Mean and variance of a random variable — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMean and variance of a random variable5 Marks
Question
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
✓

Answer

We can select two positive in 6 × 5 = 30 different waysP(X = 3) = P(larger number is 3) = {(2, 3), (3, 2)} $=\frac{2}{30}$
P(X = 4) = P(larger number is 4) = {(2, 4), (4, 2), (3, 4), (4, 3)} $=\frac{4}{30}$
P(X = 5) = P(larger number is 5) = {(2, 5), (5, 2), (3, 5), (5, 3), (4, 5), (5, 4)} $=\frac{6}{30}$
P(X = 6) = P(larger number is 6) = {(2, 6), (6, 2), (3, 6), (6, 3), (4, 6), (6, 4), (5, 6), (6, 5)} $=\frac{8}{30}$
P(X = 7) = P(larger number is 7) = {(2, 7), (7, 2), (3, 7), (7, 3), (4, 7), (7, 4), (5, 7), (7, 5), (6, 7), (7, 6)} $=\frac{10}{30}$
Thus, the probability distribution of random variable X is,
$\text{x}_\text{i}$ $\text{p}_\text{i}$ $\text{x}_\text{i}\text{p}_\text{i}$ $\text{x}_\text{i}^2\text{p}_\text{i}$
$3$ $\frac{2}{30}$ $\frac{6}{30}$ $\frac{18}{30}$
$4$ $\frac{4}{30}$ $\frac{16}{30}$ $\frac{64}{30}$
$5$ $\frac{6}{30}$ $\frac{30}{30}$ $\frac{150}{30}$
$6$ $\frac{8}{30}$ $\frac{48}{30}$ $\frac{288}{30}$
$7$ $\frac{10}{30}$ $\frac{70}{30}$ $\frac{490}{30}$
    $\sum\text{x}_\text{i}\text{p}_\text{i}=\frac{17}{3}$ $\sum\text{x}_\text{i}\text{p}_\text{i}^2=\frac{101}{3}$
Mean $=\sum\text{x}_\text{i}\text{p}_\text{i}=\frac{17}{3}$
Variance $=\sum\text{x}_\text{i}\text{p}_\text{i}-\big(\sum\text{x}_\text{i}\text{p}_\text{i}\big)^2=\frac{101}{3}-\Big(\frac{17}{3}\Big)=\frac{14}{9}$

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

More "5 Marks Questions" from Mean and variance of a random variable→Mean and variance of a random variable — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the vector equation of the plane passing through the intersection of the planes $\vec{\text{r}}.\Big(2\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}\Big)=7,\ \vec{\text{r}}.\Big(2\hat{\text{i}}+5\hat{\text{j}}+3\hat{\text{k}}\Big)=9$ and through the point (2, 1, 3).
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\sin\text{x}}+\big(\sin\text{x}\big)^\text{x}$
→
Evaluate:$\int\limits_0^{\pi/2}\frac{\text{x + sin x}}{\text{1 + cosx}}\text{dx}$.
→
Let $\text{A}=\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix}$ Use the principle of mathematical induction to show that
$\text{A}^\text{n}=\begin{bmatrix}1&\text{n}&\frac{\text{n}(\text{n}+1)}{2}\\0&1&\text{n}\\0&0&1\end{bmatrix}$ for every positive integer n.
→
Differentiate the following w.r.t. x:
$(\text{x}+1)^2(\text{x}+2)^3(\text{x}+3)^4$
→
If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$.
→
Using differentials, find the approximate values of the following:
$(1.999)^5$
→
$\text{Let } \vec{\text a} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}}, \vec{\text{b}} = \hat{\text{i}} \text{ and } \vec{\text{c}} = \text{c}_{1} \hat{\text{i}} + \text{c}_{2} \hat{\text{j}} + \text{c}_{3} \hat{\text{k}}, \text{then}$
  1. Let $c_1 = 1$ and $c_2 = 2,$ find $c_3$ which makes $\vec{\text{a}}, \vec{\text{b}} \text{ and }\vec{\text{c}} \text{ coplanar.}$
  2. If $c_2 = –1$ and $c_3 = 1,$ show that no value of $c_1 $ can make $\vec{\text{a}}, \vec{\text{b}} \text{ and } \vec{\text{c}} \text{ coplanar}.$
→
Evalute the following integrals:
$\int\frac{\text{cosec x}}{\log\tan\frac{\text{x}}{2}}\text{dx}$
→
An unbiased coin is tossed 4 times, Find the mean and variance of the number of heads obtained.
→