Page 58 - DMTH404_STATISTICS

Statistics

Notes (i) p(X )  0  i = 1, 2, ...... n and
i

n
(ii) å p   1X 
i
i 1
In a similar way, the distribution of a continuous random variable is called a Continuous
Probability Distribution and the corresponding probability function p(X) is termed as the
Probability Density Function. The conditions for any function of a continuous variable to serve
as a probability density function are:
(i) p(X)  0  real values of X, and

¥
(ii) -¥ ò p X 1
 dX 
Remarks:

1. When X is a continuous random variable, there are an infinite number of points in the
sample space and thus, the probability that X takes a particular value is always defined to
be zero even though the event is not regarded as impossible. Hence, we always measure
the probability of a continuous random variable lying in an interval.

2. The concept of a probability distribution is not new. In fact it is another way of representing
a frequency distribution. Using statistical definition, we can treat the relative frequencies
of various values of the random variable as the probabilities.

Example 2: Two unbiased die are thrown. Let the random variable X denote the sum of
points obtained. Construct the probability distribution of X.
Solution.
The possible values of the random variable are:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

The probabilities of various values of X are shown in the following table:
Probability Distribution of X

X 2 3 4 5 6 7 8 9 10 11 12 Total
p X a f 1 2 3 4 5 6 5 4 3 2 1 1
36 36 36 36 36 36 36 36 36 36 36

Example 3: Three marbles are drawn at random from a bag containing 4 red and 2 white
marbles. If the random variable X denotes the number of red marbles drawn, construct the
probability distribution of X.
Solution.

The given random variable can take 3 possible values, i.e., 1, 2 and 3. Thus, we can compute the
probabilities of various values of the random variable as given below:

4 C ´ 2 C 4
P(X = 1, i.e., 1R and 2 W marbles are drawn)  1 2 
6
C 20
3
4 C ´ 2 C 12
P(X = 2, i.e., 2R and 1W marbles are drawn)  2 1 
6
C 20
3

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