If the probability that the random variable X takes the value x i… - Vidyadip

MCQ

If the probability that the random variable $X$ takes the value x is given by $\mathrm{P}(\mathrm{X}=\mathrm{x})=\mathrm{k}(\mathrm{x}+1) 3^{-\mathrm{x}}$, $\mathrm{x}=0,1,2,3 \ldots \ldots$, where k is a constant, then $\mathrm{P}(\mathrm{X} \geq 3)$ is equal to

A
$\frac{7}{27}$
B
$\frac{4}{9}$
C
$\frac{8}{27}$
D
$\frac{1}{9}$

✓

Answer

D. $\frac{1}{9}$
$\sum_{x=0}^{\infty} k(x+1) 3^{-x}=1$
$\Rightarrow \frac{1}{\mathrm{k}}=1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+ \quad\quad...(i)$
$\frac{1}{3 \mathrm{k}}=\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\quad\quad\quad \ldots(ii)$
(i)- (ii) $\Rightarrow \frac{1}{\mathrm{k}}-\frac{1}{3 \mathrm{k}}=1+\frac{1}{3}+\frac{1}{3^{2}}+\ldots$
$\Rightarrow \mathrm{k}=\frac{4}{9}$
$P(x \geq 3)=1-P(x=0)-P(x=1)-P(x=2)$
$=1-\mathrm{k}\left(1+\frac{2}{3}+\frac{3}{9}\right)=\frac{1}{9}$

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 "SECTION - A [MATHS - MCQ]" from JEE Main 3-April-2025 Paper - Shift 2→JEE Main 3-April-2025 Paper - Shift 2 — all question types→JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If for some $\alpha$ and $\beta$ in $R,$ the intersection of the following three planes $x+4 y-2 z=1$ ; $x+7 y-5 z=\beta$ ; $x+5 y+\alpha z=5$ is a line in $\mathrm{R}^{3},$ then $\alpha+\beta$ is equal to

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event. Given below are two statements :

$(S1)$ : If $P ( A )=0$, then $A =\phi$

$( S 2)$ : If $P ( A )=$, then $A =\Omega$

Then

Tangents are drawn from points onthe circle $x^2 + y^2 = 49$ to the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{24}} = 1$ angle between the tangents is

The equation of the common tangent of the parabolas ${x^2} = 108y$ and ${y^2} = 32x$, is

The solution of the inequallity $(cot^{-1}x)^2 -5cot^{-1}x + 6 > 0$ is :-

A circle with centre $(a, b)$ passes through the origin. The equation of the tangent to the circle at the origin is

The eccentricity of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1$ is

If $y^{1 / 4}+y^{-1 / 4}=2 x$, and $\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\alpha x \frac{d y}{d x}+\beta y=0$ then $|\alpha-\beta|$ is equal to ...... .

Consider the following system of questions $\alpha x+2 y+z=1$ ; $2 \alpha x+3 y+z=1$ ; $3 x+\alpha y+2 z=\beta$ . For some $\alpha, \beta \in R$. Then which of the following is NOT correct.

Which of the following statement is true for the function $f(x)\, = \left[ \begin{array}{l}\sqrt x \,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\, \ge \,1\\{x^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\, \le x\, \le 1\\\frac{{{x^3}}}{3}\, - \,4x\,\,\,\,\,x\,\, < \,0\end{array} \right.$