If the mean and variance of a random variable X having a binomial… - Vidyadip

Question

If the mean and variance of a random variable X having a binomial distribution are 4 and 2 respectively, find P (X = 1).

✓

Answer

Let n and p be the parmeters of binomial distribution,
Given,
$\text{Mean = np}=4$
$\text{Variance = npq}=2$
Dividing equation (2) by (1),
$\frac{\text{npq}}{\text{np}}=\frac{2}{4}$
$\text{q}=\frac{1}{2}$.
$\text{p}=1-\frac{1}{2}$
$\text{p}=\frac{1}{2}$
Put the value of p in equation (1),
$\text{np}=4$
$\text{n}\big(\frac{1}{2}\big)=4$
$\text{n}=8$
Hence, binomial distribution is given by
$\text{P(X = r})=\text{ }^{\text{n}}\text{c}_{\text{r}}\text{p}^{\text{r}}\text{q}^{\text{n}-\text{r}}$
$\text{P(X= r})\text{ }^8\text{c}_{\text{r}}\big(\frac{1}{2}\big)^{\text{r}}\big(\frac{1}{2}\big)^{8-\text{r}}$
$\text{P(X=}1)$
$=\text{ }^8\text{c}_{1}\big(\frac{1}{2}\big)^{1}\big(\frac{1}{2}\big)^{8-1}$
$=8\big(\frac{1}{2}\big)^8$
$=\big(\frac{1}{2}\big)^5$
$=\frac{1}{32}$
$\text{P(X}=1)=\frac{1}{32}$

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 Binomial Distribution→Binomial Distribution — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Write the number of points where f(x) = |x| + |x − 1| is continuous but not differentiable.

The probability distribution of a random variable x is given as under:
$\text{P}(\text{X}=\text{x})=\begin{cases}\text{k}\text{x}^2 & \text{for}\text{ x}= 1,2,3\\2\text{kx} & \text{for}\text{ x } =4,5,6\\0&\text{otherwise} \end{cases}$
where k is a constant. Calculate

$\text{E}(\text{X})$
$\text{E}(3\text{X}^2)$
$\text{P}(\text{X}\geq4)$

Differentiate $\tan^{-1} \bigg(\frac{\sqrt{1 + x^{2} - 1}}{x}\bigg)$ w.r.t. $\sin^{-1} \frac{2x}{1 + x^{2}}, \text{ if } x \in (-1, 1)$

A bag A contains $5$ white and $6$ black balls. Another bag B contains $4$ white and $3$ black balls. A ball is transferred from bag A to the bag B and then a ball is taken out of the second bag. Find the probability of this ball being black.

Find the equation of the lines joining the following pairs of vertices and then find the shortest distance between the lines(1) (0, 0, 0) and (1, 0, 2)
(2) (1, 3, 0) and (0, 3, 0)

Determine whether the relation R defined on the set $\Re$ of all real numbers as R =$(\text{a,b) : a, b} \in \Re$ and $\text{a - b} + \sqrt{3} \in \text{S},$where S is the set of all irrational numbers, is reflexive, symmetric and transitive.

$\text{Let}\ \vec{\text{a}}=4\hat{\text{i}}+5\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{c}}\ \text{and }\vec{\text{b}}\ \text{and}\ \vec{\text{d}}\cdot\vec{\text{a}}=21.$

If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}},\ 4\vec{\text{a}}+3\vec{\text{b}},\ 10\vec{\text{a}}+7\vec{\text{b}}-2\vec{\text{c}}$

Prove the following Exercise:
$\int^{3}\limits_{1}\frac{\text{dx}}{\text{x}^{2}(\text{x}+1)}=\frac{2}{3}+\log\frac{2}{3}$

If $\text{A}=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix},$ find $A^2.$