STA301 Final Term Highlighted Handouts by Masters

LECTURE NO. 23

Graphical Representation of the Distribution Function of a Discrete Random Variable

Mathematical Expectation

Mean, Variance and Moments of a Discrete Probability Distribution

Properties of Expected Values

First, let us consider the concept of the DISTRIBUTION FUNCTION of a discrete random variable.

DISTRIBUTION FUNCTION

The distribution function of a random variable X, denoted by F(x), is defined by F(x) = P(X < x). The function F(x) gives the probability of the event that X takes a value LESS THAN OR EQUAL TO a specified value x. The distribution function is abbreviated to d.f. and is also called the cumulative distribution function (cdf) as it is the cumulative probability function of the random variable X from the smallest value up to a specific value x.

EXAMPLE

Find the probability distribution and distribution function for the number of heads when 3 balanced coins are tossed. Depict both the probability distribution and the distribution function graphically. Since the coins are balanced, therefore the equally probable sample space for this experiment is

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.

Let X be the random variable that denotes the number of heads. Then the values of X are 0, 1, 2 and 3. And their probabilities are:

f (0) = P(X = 0) = P [{TTT}] = 1/8 f(1) = P (X = 1) = P [{HTT, THT, TTH}] = 3/8 f(2) = P (X = 2) = P [{HHT, HTH, THH}] = 3/8 f(2) = P (X = 3) = P [{HHH}] = 1/8

Expressing the above information in the tabular form, we obtain the desired probability distribution of X as follows:

Number of Heads · (xi) · f(xi) · Total

The line chart of the above probability distribution is as follows:

In order to obtain the distribution function of this random variable, we compute the cumulative probabilities as follows:

Hence the desired distribution function is

Why has the distribution function been expressed in this manner? The answer to this question is:

If x < 0, we have P(X < x) = 0, the reason being that it is not possible for our random variable X to assume value less than zero.(The minimum number of heads that we can have in tossing three coins is zero.) If 0 < x < 1, we note that it is not possible for our random variable X to assume any value between zero and one. (We will have no head or one head but we will NOT have 1/3 heads or 2/5 heads!) Hence, the probabilities of all such values will be zero, and hence we will obtain a situation which can be explained through the following table:

X · f(x) · Number of · Heads · (xi) · f(xi) · F(xi) · ,1 · ,0 · x · for · x · for · x · for · x · for · x · for · x · F

The above table clearly shows that the probability that X is LESS THAN any value lying between zero and 0.9999… will be equal to the probability of X = 0 i.e. For 0 < x < 1,

Similarly,

For 1 < x < 2, we have

For 2 < x < 3, we have

And, finally, for x > 3, we have

Hence, the graph of the DISTRIBUTION FUNCTION is as follows:

Number of · Heads · (xi) · f(xi) · F(xi) · 0.2 · 0.4 · 0.6 · 0.8 · 0) · P(X · x) · P(X · X · P · X · P · x · X · P · X · P · X · P · X · P · x · X · P · .1 · X · P · X · P · X · P · X · P · x · X · P