1. Discrete Random Variables and Probability Distributions
Chapter 5
Discrete Random Variables and Probability Distributions

2. Random Variables
A random variable is a variable that takes on numerical values determined by the outcome of a random experiment.

3. Discrete Random Variables
A random variable is discrete if it can take on no more than a countable number of values.

4. Discrete Random Variables (Examples)
The number of defective items in a sample of twenty items taken from a large shipment.
The number of customers arriving at a check-out counter in an hour.
The number of errors detected in a corporation’s accounts.
The number of claims on a medical insurance policy in a particular year.

5. Continuous Random Variables
A random variable is continuous if it can take any value in an interval.

6. Continuous Random Variables (Examples)
The income in a year for a family.
The amount of oil imported into the U.S. in a particular month.
The change in the price of a share of IBM common stock in a month.
The time that elapses between the installation of a new computer and its failure.
The percentage of impurity in a batch of chemicals.

7. Discrete Probability Distributions
The probability distribution function (DPF), P(x), of a discrete random variable expresses the probability that X takes the value x, as a function of x. That is

8. Discrete Probability Distributions
Graph the probability distribution function for the roll of a single six-sided die.
P(x)
1/6 1 2 3 4 5 6 x
Figure 5.1

9. Required Properties of Probability Distribution Functions of Discrete Random Variables
Let X be a discrete random variable with probability distribution function, P(x). Then
P(x)  0 for any value of x
The individual probabilities sum to 1; that is
Where the notation indicates summation over all possible values x.

10. Cumulative Probability Function
The cumulative probability function, F(x0), of a random variable X expresses the probability that X does not exceed the value x0, as a function of x0. That is
Where the function is evaluated at all values x0

11. Derived Relationship Between Probability and Cumulative Probability Function
Let X be a random variable with probability function P(x) and cumulative probability function F(x0). Then it can be shown that
Where the notation implies that summation is over all possible values x that are less than or equal to x0.

12. 0  F(x0)  1 for every number x0
Derived Properties of Cumulative Probability Functions for Discrete Random Variables
Let X be a discrete random variable with a cumulative probability function, F(x0). Then we can show that
0  F(x0)  1 for every number x0
If x0 and x1 are two numbers with x0 < x1, then F(x0)  F(x1)

13. Expected Value
The expected value, E(X), of a discrete random variable X is defined
Where the notation indicates that summation extends over all possible values x.
The expected value of a random variable is called its mean and is denoted x.

14. Variance and Standard Deviation
Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X - )2, is called the variance, denoted 2x and is given by
The standard deviation, x , is the positive square root of the variance.

15. Variance (Alternative Formula)
The variance of a discrete random variable X can be expressed as

16. Expected Value and Variance for Discrete Random Variable Using Microsoft Excel (Figure 5.4)

17. Bernoulli Distribution
A Bernoulli distribution arises from a random experiment which can give rise to just two possible outcomes. These outcomes are usually labeled as either “success” or “failure.” If  denotes the probability of a success and the probability of a failure is (1 -  ), the the Bernoulli probability function is

18. Mean and Variance of a Bernoulli Random Variable
The mean is:
And the variance is:

19. Sequences of x Successes in n Trials
The number of sequences with x successes in n independent trials is:
Where n! = n x (x – 1) x (n – 2) x x 1 and 0! = 1.

20. Binomial Distribution
Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that  is the probability of a success resulting in a single trial. If n independent trials are carried out, the distribution of the resulting number of successes “x” is called the binomial distribution. Its probability distribution function for the binomial random variable X = x is:
P(x successes in n independent trials)=
for x = 0, 1, , n

21. Mean and Variance of a Binomial Probability Distribution
Let X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with mean,
and variance,

22. Binomial Probabilities - An Example – (Example 5.7)
An insurance broker, Shirley Ferguson, has five contracts, and she believes that for each contract, the probability of making a sale is 0.40.
What is the probability that she makes at most one sale?
P(at most one sale) = P(X  1) = P(X = 0) + P(X = 1)
= = 0.337

23. Binomial Probabilities, n = 100,  =0.40 (Figure 5.10)

24. Hypergeometric Distribution
Suppose that a random sample of n objects is chosen from a group of N objects, S of which are successes. The distribution of the number of X successes in the sample is called the hypergeometric distribution. Its probability function is:
Where x can take integer values ranging from the larger of 0 and [n-(N-S)] to the smaller of n and S.

25. Poisson Probability Distribution
Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson probability distribution are:
The probability of an occurrence of an event is constant for all subintervals.
There can be no more than one occurrence in each subinterval.
Occurrences are independent; that is, the number of occurrences in any non-overlapping intervals in independent of one another.

26. Poisson Probability Distribution
The random variable X is said to follow the Poisson probability distribution if it has the probability function:
where
P(x) = the probability of x successes over a given period of time or space, given 
 = the expected number of successes per time or space unit;  > 0
e = (the base for natural logarithms)

27. Poisson Probability Distribution
The mean and variance of the Poisson probability distribution are:

28. Partial Poisson Probabilities for  = 0
Partial Poisson Probabilities for  = 0.03 Obtained Using Microsoft Excel PHStat (Figure 5.14)

29. Poisson Approximation to the Binomial Distribution
Let X be the number of successes resulting from n independent trials, each with a probability of success, . The distribution of the number of successes X is binomial, with mean n. If the number of trials n is large and n is of only moderate size (preferably n  7), this distribution can be approximated by the Poisson distribution with  = n. The probability function of the approximating distribution is then:

30. Joint Probability Functions
Let X and Y be a pair of discrete random variables. Their joint probability function expresses the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y. The notation used is P(x, y) so,

31. Joint Probability Functions
Let X and Y be a pair of jointly distributed random variables. In this context the probability function of the random variable X is called its marginal probability function and is obtained by summing the joint probabilities over all possible values; that is,
Similarly, the marginal probability function of
the random variable Y is

32. Properties of Joint Probability Functions
Let X and Y be discrete random variables with joint probability function P(x,y). Then
P(x,y)  0 for any pair of values x and y
The sum of the joint probabilities P(x, y) over all possible values must be 1.

33. Conditional Probability Functions
Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X. This is denoted P(y|x), and so by the definition of conditional probability:
Similarly, the conditional probability function of X, given Y = y is:

34. Independence of Jointly Distributed Random Variables
The jointly distributed random variables X and Y are said to be independent if and only if their joint probability function is the product of their marginal probability functions, that is, if and only if
And k random variables are independent if and only if

35. Expected Value Function of Jointly Distributed Random Variables
Let X and Y be a pair of discrete random variables with joint probability function P(x, y). The expectation of any function g(x, y) of these random variables is defined as:

36. Stock Returns, Marginal Probability, Mean, Variance (Example 5.16)
Y Return
X Return 0% 5%
10%
15%
0.0625
Table 5.6

37. For discrete random variables An equivalent expression is
Covariance
Let X be a random variable with mean X , and let Y be a random variable with mean, Y . The expected value of (X - X )(Y - Y ) is called the covariance between X and Y, denoted Cov(X, Y).
For discrete random variables
An equivalent expression is

38. Correlation
Let X and Y be jointly distributed random variables. The correlation between X and Y is:

39. Covariance and Statistical Independence
If two random variables are statistically independent, the covariance between them is 0. However, the converse is not necessarily true.

40. Portfolio Analysis
The random variable X is the price for stock A and the random variable Y is the price for stock B. The market value, W, for the portfolio is given by the linear function,
Where, a, is the number of shares of stock A and, b, is the number of shares of stock B.

41. Portfolio Analysis The mean value for W is, The variance for W is,
or using the correlation,