Let X be a random variable such that the probability function of a distribution is given byPX - 0 - 12- PX - j - 13j j - 1- 2- 3- Then the mean of the distribution and PX is positive and even respectively are

Mean of distribution (M) = ∑ x_i P_i ⇒ M=0 ( 12 ) + 1 ( 13 ) + 2 ( 1 3^2) + 3 ( 13^3) + …∞ ⇒ nbsp;M = 13 + 2 ( 13^2) +3 ( 1 3^3) +…∞ ⇒ nbsp; M3 = 13^ ...

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Standard XII

Mathematics

Coefficient of Variation

Let X be a ra...

Question

Let $X$ be a random variable such that the probability function of a distribution is given by $P (X = 0) = \frac{1}{2}, P (X = j) = \frac{1}{3^{j}} (j = 1, 2, 3, \dots, \infty)$ . Then the mean of the distribution and $P (X is positive and even)$ respectively are

\frac{3}{4}

and

\frac{1}{9}

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\frac{3}{4}

and

\frac{1}{16}

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\frac{3}{4}

and

\frac{1}{8}

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\frac{3}{8}

and

\frac{1}{8}

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Solution

The correct option is C $\frac{3}{4}$ and $\frac{1}{8}$
Mean of distribution $(M)$
$= \sum x_{i} P_{i}$
$\Rightarrow M = 0 (\frac{1}{2}) + 1 (\frac{1}{3}) + 2 (\frac{1}{3^{2}}) + 3 (\frac{1}{3^{3}}) + \dots \infty$
$\Rightarrow M = \frac{1}{3} + 2 (\frac{1}{3^{2}}) + 3 (\frac{1}{3^{3}}) + \dots \infty$
$\Rightarrow \frac{M}{3} = \frac{1}{3^{2}} + 2 (\frac{1}{3^{3}}) + \dots \infty$
$∴ M - \frac{M}{3} = \frac{1}{3} + \frac{1}{3^{2}} + \frac{1}{3^{3}} + \dots \infty$
$\Rightarrow \frac{2 M}{3} = \frac{\frac{1}{3}}{\frac{2}{3}} \Rightarrow \frac{2 M}{3} = \frac{1}{2}$
$∴$ Mean $= \frac{3}{4}$

$P (X is positive and even)$
$= \frac{1}{3^{2}} + \frac{1}{3^{4}} + \frac{1}{3^{6}} + \dots$
$= \frac{\frac{1}{3^{2}}}{1 - \frac{1}{3^{2}}} = \frac{1}{8}$

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Let X be a random variable such that the probability function of a distribution is given byP(X=0)=12, P(X=j)=13j (j=1,2,3,…,∞). Then the mean of the distribution and P(X is positive and even) respectively are

The correct option is C 34 and 18Mean of distribution (M) =∑xiPi ⇒M=0(12)+1(13)+2(132)+3(133)+…∞ ⇒M=13+2(132)+3(133)+…∞ ⇒M3=132+2(133)+…∞ ∴M−M3=13+132+133+…∞ ⇒2M3=1323⇒2M3=12 ∴ Mean =34 P(X is positive and even) =132+134+136+… =1321−132=18

Let $X$ be a random variable such that the probability function of a distribution is given by $P (X = 0) = \frac{1}{2}, P (X = j) = \frac{1}{3^{j}} (j = 1, 2, 3, \dots, \infty)$ . Then the mean of the distribution and $P (X is positive and even)$ respectively are