If a variable can be assigned any value between two values then it a continuous variable. If only a fixed number of values can be taken such as 2, 3, 4, 5, then it is known as discrete variable. The mean of a discrete probability distribution can be written as:
$\mu =\sum_{x=i}^{n}xP(x)$
The standard deviation of a probability distribution can be given as:
$\sigma = \sqrt{(X-\mu )^{2}P(X)}$
Discrete probability distributions are of various forms. Prominent of them are binomial distribution, hypergeometric distribution, poisson's distribution, multinomial distribution.

Word Problems

Problem 1:

Find the mean value of variable X from the given probability distribution.

 X  0 1
 P(X)  0.3  0.2  0.5


Mean of variable X,

$\mu =\sum_{x=i}^{n}XP(X)$

$\mu = 0\times 0.3 + 1\times 0.2+2\times 0.5=1.2$

Hence, mean value of this probability distribution is 1.2.
Problem 2:

In a draw, only numbers 3, 5 and 6 can come and given is the probability distribution for these numbers. Find the mean of the numbers obtained in the draw.

3  5  6
P(X)  0.6  0.3  0.1


Let the numbers obtained be X. The mean of X can be derived using the formula:

$\mu =\sum_{x=i}^{n}XP(X)$

$\mu = 3\times 0.6 + 5\times 0.3+6\times 0.1= 3.9$

Mean of numbers got in the draw is 3.9.
Problem 3:

For a variable X, the mean is 1.1 and probability distribution is given below.

 1  2
 0.2  0.5 0.3

Find the standard deviation of X.


The standard deviation of X can be obtained using the formula,

$\sigma = \sqrt{(X-\mu )^{2}P(X)}$

$\sigma = \sqrt{(0-1.1 )^{2}\times 0.2+(1-1.1 )^{2}\times 0.5+(2-1.1 )^{2}\times 0.3}$

$\sigma = \sqrt{0.242 + 0.005 + 0.243} = \sqrt 0.49$

$\sigma = 0.7$

Standard deviation is 0.7.
Problem 4:

Make the probability distribution table for getting a head if two coins are tossed together.


Sample space, S = {HH, HT, TH, TT}

Numbers of heads can be 0, 1 or 2.

The table for probability distribution will be,

 0  1  2
 P(X)  $\frac{1}{4}$  $\frac{2}{4}$  $\frac{1}{4}$