Download Discrete Probability Distribution

Discrete Probability Distributions
Random Variables and Their Probability Distributions
A random variable is a real-valued function whose domain is a sample
space.
A random variable X is said to be discrete if it can take on only a finite
number, or a countably infinite number, of possible values x.
The function p(x) is called the probability function of X, provided
1. P(X = x) = p(x)  0 for all values of x
2.
Example (Table 5.1) of the probability distribution of the number of closed
relays of an electrical circuit.
Table 5.1 Probability Distribution of Number of Closed Relays
x
P(x)
0
0.04
1
0.32
2
0.64
Total
1.00
Figure 5.2 Probability distribution of number of closed relays
The distribution function F(b) for a random variable X is defined as
F(b) = P(X  b).
The distribution function is also known as the cumulative distribution
function (cdf).
Example: The random variable X, denoting the number of relays closing
properly of the electrical circuit has the probability distribution given
below:
P(X = 0)
0.04
P(X = 1)
0.32
P(X = 2)
0.64
Note that P(X  1.5) = P(X =0) + P(X = 1) = 0.04 + 0.32 = 0.36
The distribution function for this random variable (Figure 5.4) has the
form
Figure 5.3 A distribution function for a discrete random variable
Expected Values of Random Variables
The expected value of a discrete random variable X having probability
function p(x) is given by
The sum is over all values of x for which p(x) > 0.
The variance of a random variable X with expected value  is given by
The standard deviation of a random variable X is the square root of the
variance given by
Example:
The manager of a stockroom in a factory knows from her study of records
that X, the daily demand (number of times used per day) for a certain
tool, has the following probability distribution:
Demand
Probability
0
0.1
1
0.5
2
0.4
Find the expected daily demand for the tool and the variance.
Solution:
From the definition of expected value,
E(X) =
Therefore, the tool is used an average of 1.3 times per day.
The variance is
V(X) =
For any random variable X and constants a and b.
1. E(aX + b) = aE(X) + b
2. V(aX + b) = a2V(X)
Example: In the previous example, suppose it costs the factory $100
each time the tool is used. Find the mean and variance of the daily costs
for use of this tool.
Solution:
Recall that X = the daily demand. Then the daily cost of using this tool is
100X. Hence
E(100X) = 100 E(X) = 100(1.3) = 130
and
V(100X) = 1002V(X) = 10,000(0.41) = 4,100
The factory should budget $130 per day to cover the cost of using the
tool. The standard deviation of daily cost is $64.03.
Example:
The operators of certain machinery are required to take a proficiency test.
A national testing agency gives this test nationwide. The possible scores
on the test are 0, 1, 2, 3, or 4. The test brochure published by the agency
gave the following information about the scores and the corresponding
probabilities:
Score
Probability
0
0.1
1
0.2
2
0.4
3
0.2
Compute the mean score and the standard deviation.
Solution:
The mean score or the expected score is
Since,
The variance can be computed as
The standard deviation of scores is
and
4
0.1