Tutorial Discrete Random Variable


Discrete Random Variable


   


A random variable is called discrete if it can assume finite number of values. If two bulbs are selected from a certain lot, the number of defective bulbs may be 0, 1 or 2. The range of the variable is from 0 to 2 and random variable can take some selected values in this range. The number of defective bulbs cannot be 1.1 or or 3 etc. This random variable can take only the specific values which are 0, 1 and 2. When two dice are rolled the total on the two dice will be 2, 3, …, 12. The total on the two dice is a discrete random variable.

Discrete Probability Distribution:
Suppose a discrete random variable can assume the values with corresponding probabilities . The set of ordered pairs is called the probability distribution or probability function of random variable . A probability distribution can be presented in a tabular form showing the values of the random variable and the corresponding probabilities donated by or . The above information can be collected in the form of a table below called the probability distribution of the random variable . The probability that random variable will take the value is denoted by where.

Values of random variable

Probability

The probability of some interval can be calculated by adding the probabilities of all points in the interval. For example.
This type of addition will not be possible in continuous random variable for finding the probability of an interval. Therein we shall use the integral calculus for finding the probability of an interval.
The probability distribution can also be described in the form of an equation for with a list of possible values of the random variable. Some probability distributions in the form of equations are

(i) for
For each value of the probability is 1/6. It is the probability distribution when a fair die is rolled.

(ii) for

Example:

A digit is selected from the first 8 natural numbers. Write the probability distribution of where is the number of factors (divisors) of digits.

Solution:

It is assumed here that the probability of selection for each digit is . We can write:

Digit

Factors

Number of factors

Probability

1

1

1

1/8

2

1, 2

2

1/8

3

1, 3

2

1/8

4

1, 2, 4

3

1/8

5

1, 5

2

1/8

6

1, 2, 3, 6

4

1/8

7

1, 7

2

1/8

8

1, 2, 4, 8

4

1/8

The information in the above table can be summarized as below in the form of table of probability distribution.

Number of factors r.v.

1

2

3

4

Total

;

1/8

4/8

1/8

2/8

1

The probability of is 4/8 which has been obtained by adding 1/8 four times. Similarly the probability of 4 is 2/8 which has been obtained by adding 1/8 two times.

Example:
A factory is producing bulbs out of which 25% are defective. Two bulbs are selected from this factory for inspection. Write the probability distribution of number of defective bulbs.

Solution:
Let denotes the number of defective bulbs. Let denote the defective and denote the good bulbs.

Here
; and

It is assumed here that the two bulbs are selected independently because there is very large number of bulbs in the factory. According to multiplication law of independent events, we have




The possible outcomes, the random variable and the corresponding probabilities are put in the following tables:

Outcomes
(S)

Number of defectives
(r.v.)

Probability

2

1

1

0

The above information can be collected as below in the form of a probability distribution of the random variable.

Random Variable

0

1

2

9/16

6/16

1/16




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