Name: Probability Distribution of a Random Variable
Uploaded: 2019-07-19
Duration: 3 min 52 s
Description: If the sum of the probabilities in the distribution is one, then it is called a probability distribution. Watch the video to know how this fact is used in ma...

Question

Three cards are drawn successively with replacement from a well – shuffled deck of

52

cards. A random variable

X

denotes the number of hearts in the three cards drawn. Find the mean and variance of

X

.

Accepted Answer

Three cards are drawn successively with replacement from a well – shuffled deck of $X$ cards.

Let $X$ be the number of hearts drawn.

Number of hearts in $52$ cards is $13$

$P (0) = \frac{39}{52} \times \frac{39}{52} \times \frac{39}{52} = \frac{27}{64}$

$P (1) = {}^{2}C_{1} \times \frac{13}{52} \times \frac{39}{52} \times \frac{39}{52} = \frac{27}{64}$

$P (2) = {}^{3}C_{2} \times \frac{13}{52} \times \frac{13}{52} \times \frac{39}{52} = \frac{9}{64}$

$P (3) = \frac{13}{52} \times \frac{13}{52} \times \frac{13}{52} = \frac{1}{64}$

The probability distribution table is as follows,

$x_{i}$ $0$ $1$ $2$ $3$

$p_{i}$ $\frac{27}{64}$ $\frac{27}{64}$ $\frac{9}{64}$ $\frac{1}{64}$

Mean $= E (X) = \sum_{i = 0}^{3} x_{i} p (x_{i}) = x_{0} p (x_{0}) + x_{1} p (x_{1}) + x_{2} p (x_{2}) + x_{3} p (x_{3})$

$= 0 (\frac{27}{64}) + 1 (\frac{27}{64}) + 2 (\frac{9}{64}) + 3 (\frac{1}{64})$

$= 0 + \frac{27}{64} + \frac{18}{64} + \frac{3}{64}$

$= \frac{48}{64} = \frac{3}{4}$

Variance $= E (X^{2}) - {[E (X)]}^{2}$

Consider, ${[E (X)]}^{2} = {(\frac{3}{4})}^{2} = \frac{9}{16}$ and

$E (X^{2}) = \sum_{i = 0}^{3} {(x_{i})}^{2} \cdot p (x_{i})$

$= (0)^{2} (\frac{27}{64}) + (1)^{2} (\frac{27}{64}) + (2)^{2} (\frac{9}{64}) + (3)^{2} (\frac{1}{64})$

$= 0 + \frac{27}{64} + \frac{36}{64} + \frac{9}{64} = \frac{72}{64} = \frac{9}{8}$

Variance $= \frac{9}{8} - \frac{9}{16} = \frac{18 - 9}{16} = \frac{9}{16}$

Three cards are drawn successively with replacement from a well – shuffled deck of $52$ cards. A random variable $X$ denotes the number of hearts in the three cards drawn. Find the mean and variance of $X$ .

Important Questions on Probability Distribution

Learn and Prepare for any exam you want !

Three cards are drawn successively with replacement from a well – shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Find the mean and variance of X.

Important Questions on Probability Distribution

Learn and Prepare for any exam you want !

Three cards are drawn successively with replacement from a well – shuffled deck of $52$ cards. A random variable $X$ denotes the number of hearts in the three cards drawn. Find the mean and variance of $X$ .