STA301 Lectures 23-45 by Masters

STA301 highlighted handouts final term by masters

Bs Computer science (Virtual University of Pakistan)

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STA301 highlighted handouts final term by masters

Bs Computer science (Virtual University of Pakistan)

Scan to open on Studocu

Studocu is not sponsored or endorsed by any college or university

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