Probability mass function (or briefly pmf) of a discrete random variable `X`, symbolized as `p(x)`, is defined as:

`p(x)=P(X=x)`(1)

Assume that `x` can take values `x_1,\ x_2,\ ...\ ,\ x_i,\ ...\ ,\ x_n` , then:

• if `x=x_i` : `p(x_i)=P(X=x_i)!=0`
• else `p(x)=P(X=x)=0`

Characteristics of probability mass function

From the discussion above, we can note that:

• `0<=p(x)<=1`
• `sum_(i=1)^n p(x_i)=1`

Example 1 : Rolling an homogeneous die. Define `X` as the number of dots (pips) on the upper face. Then:

• `p(2)=P(X=2)=1//6`
• `p(5)=P(X=5)=1//6`
• `p(1,25)=P(X=1,25)=0`
• `p(8)=P(X=8)=0`

We can use Table 1 to show the probability mass function.

Table 1 Probability mass function of `X`

`x`	1	2	3	4	5	6
`p(x)`	1/6	1/6	1/6	1/6	1/6	1/6

Example 2 : Rolling two homogeneous dice. Define `X` as the sum of pips on the upper faces.

As we have already discussed, `X` is a random variable defined from a sample space of 36 elements. Values of `X` are integers in the interval 2 - 12.

To calculate `p(x)`, we find the event E_`x` associates with `x`, then calculate `P("E"_x)`. For example `x=5`.

Then : E₅ = { (14), (23), (32), (41) }

So : `p(5)=P(X=5)=P("E"_5)=4//36`

Calculate similarly for other values of `X`, we obtain Table 2.

 

In general, cumulative distribution function of a random variable, symbolized as `F(x)`, is defined as:

`F(x)=P(X<=x)`(2)

It means that `F(x)` is the probability that random variable `X` is smaller than or equal to `x`.

For discrete random variable, (2) becomes:

`F(x)=sum_(x_i<=x) p(x_i)`

(3)

We have to note that :

• `x` is a real number, it means that `x` belongs to interval (`-oo,\ oo`),
• for discrete random variable, `F(x)` is a discontinuous function, with discontinuity point of the first kind.

Characteristics of cumulative distribution functions of discrete random variables

• `0<=F(x)<=1`
• `F(x)` is a non-decreasing function
• `F(-oo)=0`
• `F(oo)=1`
• `P(a<=X<=b)=F(b)-F(a)`

Example 3 : Continue Example 2 with cumulative distribution function. To calculate `F(x)`, we use formula (3). For example:

  `F(5)=p(2)+p(3)+p(4)+p(5)=1/36 + 2/36 + 3/36 + 4/36=10/36`

Calculate similarly for other values of `X`, we obtain Table 3.

 

Expected value

For discrete random variable `X`, expected value, symbolized as `mu` or `E(X)`, is defined as:

`mu=E(X)=sum_(x_i) x_i\ p(x_i)`

(4)

If we compare (4) with normalized weighted mean :

`bar x=sum_(i=1)^n w_i\ x_i`

we can recognize that expected value is normalized weighted mean, in which pmf `p(x)` plays the role of normalized weights of values.

Variance

In general, variance, denoted as `sigma^2` or `V(X)`, is defined as:

`sigma^2=V(X)=E(X-mu)^2`(5)

For discrete random variable, variance is calculated by:

`sigma^2=sum_(x_i) (x_i-mu)^2p(x_i)=sum_(x_i) x_i^2p(x_i)\ -\ mu^2`

(6)

Example 4 : We continue Example 2 with expected value and variance.

To determine expected value, we extend Table 2 to obtain Table 4.

From this result, we calculate expected value:

  `mu=sum_(x_i) x_ip(x_i)=(2+6+12+20+30+42+40+36+30+22+12)/36=252/36=7`

To determine variance, we extend Table 2 and we obtain Table 5

Therefore :

`sigma^2=sum_(x_i) (x_i-mu)^2p(x_i)=(25+32+27+16+5+0+5+16+27+32+25)/36=210/36=5,83`