Two numbers are selected at random (without replacement) from pos… - Vidyadip

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

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$

Evaluate the following intregals:
$\int\frac{3}{(1-\text{x})(1+\text{x}^2)}\ \text{dx}$

Find the cartesian and vector equations of a line which passes through the point $(1, 2, 3)$ and is parallel to the line $\frac{-\text{x}-2}{1}=\frac{\text{y}+3}{7}=\frac{2\text{z}-6}{3}.$

If f(2a - x) = -f(x), prove that $\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=0$

Find the angle between the lines whose direction ratios are proportional to a, b, c and b - c, c - a, a - b.

$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}}=\vec{\text{b }} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$

If the co$-$ordinates of the vertices of an equilateral triangle with sides of length $‘a\ ’$ are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3),$ then $\begin{vmatrix}\text{x}_1&\text{y}_1&1\\\text{x}_1&\text{y}_2&1\\\text{x}_2 &\text{y}_3&1\end{vmatrix}^2=\frac{3\text{a}^4}{4}.$

If $\text{A}=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix},$ show that $A^2 - 7A + 10I_3 = 0.$

Evaluate : $\int \frac{x^3}{(x-1)\left(x^2+1\right)} d x$