Variables

Читайте также:

Let X be a discrete random, and x be one of its possible values. The probability that the random variable X takes the value x is denoted by .

Definition:

The probability distribution function, P (x), of a discrete random variable X indicates that this variable takes the value x, as a function of x. That is

, for all values of x.

Example 1:

In the experiment of tossing a fair coin three times, let X be number of heads obtained. Determine and sketch the probability function of X.

Outcome	Value of X
HHH HHT HTH HTT THH THT TTH TTT

Solution: Table 3.1

First, X is a variable and the number of heads in three tosses of a coin can have any of the values 0, 1, 2, or 3.

We can make a list of the outcomes and the associated values of X. (Table 3.1).

Note that, for each basic outcome there is only one value of X. However, several basic outcomes may yield the same value. We identify the events

(i.e., the collections of the distinct

values of X). (Table 3.2)

Table 3.2.

Numerical value of X as an event	Composition of the event
[ X =0] [X=1] [X=2] [X=3]	{TTT} {HTT, THT, TTH} {HHT, HTH, THH} {HHH}

The model of a fair coin entails that 8 basic outcomes are equally likely, so each is assigned the probability 1/8.

The event [ X =0] has a single outcome TTT, so its probability is 1/8. Similarly, the probabilities of [ X =1], [ X =2], and [ X =3] are found to be 3/8, 3/8, and 1/8, respectively. Collecting these results, we obtain the probability distribution of X shown in table 3.3.

Table 3.3. The probability distribution of X, the

number of heads in 3 tosses of a coin.

Value of X	Probability
	1/8 3/8 3/8 1/8
Total

Remark: When summed over all possible values of X, these probabilities must add up to 1.

The graphical representation of the probability of X, the number of heads in three tosses of a coin is shown in Fig. 3.1.

In the development of the probability distribution for a discrete random variable, the following two conditions must always be satisfied:

Properties of probability function of discrete random variables:

Let X be a discrete random variable with probability P (x). Then

1. for any value x.

2. The individual probabilities sum to 1; that is , where the notation indicates summation over all possible values of x.

Another representation of discrete probability distribution is also useful.

Cumulative probability function :

The cumulative probability function, of a random variable X expresses the probability that X does not exceed the value as a function of . That is

wherethe function is evaluated of all values .

Properties of cumulative probability functions for discrete random variables:

Let X be a discrete random variable with cumulative probability function . Then we can show that

1. for every number .

2. If and are two numbers with , then .

Example2:

In the experiment of rolling a balanced die twice, let X be the minimum of the two numbers obtained. Determine and sketch the probability function and cumulative probability function of X.

Solution:

The possible values of X are 1, 2, 3, 4, 5, and 6, The sample space of this experiment consists of 36 basic outcomes. Hence the probability of any of them is 1/36. In our experiment

P (X =1)= P {(1,1)(1,2)(2,1)(1,3)(3,1)(1,4)(4,1)(1,5)(5,1)(1,6)(6,1)}=11/36

P (X =2)= P {(2,2)(2,3)(3,2)(2,4)(4,2)(2,5)(5,2)(2,6)(6,2)}=9/36

P (X =3)= P {(3,3)(3,4)(4,3)(3,5)(5,3)(3,6)(6,3)}=7/36

P (X =4)= P {(4,4)(4,5)(5,4)(4,6)(6,4)}=5/36

P (X =5) = P {(5, 5) (5, 6) (6, 5)} =3/36

P (X =6) = P {(6, 6)} =1/36

The graphical representation of is shown in Fig.3.2.

Now let us form cumulative probability function.

If is some number less than 1, X can not be less than , so

for all

If is greater than or equal to 1 but strictly less than 2, the only one number less than 2, the only way for X to be less than or equal to is if X =1. Hence

for all

If is greater than or equal to 2 but strictly less than 3, X is less than or equal to the if and only if either X =1 or X =2, so

for .

Continuing in this way we can write cumulative probability function as

The cumulative distribution function of X, , is plotted in Fig. 3.3.

It can be seen that the cumulative probability function increases in steps until the sum is 1.

Example 3:

A consumer agency surveyed all 2500 families living in a small town to collect data on the number of TV sets owned by them. The following table lists the frequency distribution of the data collected by this agency

Number of TV sets owned
Number of families

a) Construct a probability distribution table. Draw a graph of the probability distribution.

b) Calculate and draw the cumulative probability function.

c) Find the probabilities: P (X =1), P (X >1), , .

Solution:

a) In a chapter 2 we learned that the relative frequencies obtained from an experiment or a sample can be used as approximate probabilities. Using the relative frequencies, we can write the probability distribution on the discrete random variable X in the following table.