Chapter 7 : Discrete Probability
Topics covered in this snacksized chapter:
The Probability Distribution for a random variable describes how probabilities are distributed over the values of the random variable.
 We can describe a Discrete Probability Distribution with a table, graph, or equation.
Discrete outcomes are:
 Mutually exclusive (nothing in common).
 Collectively exhaustive (nothing left out).
Discrete Probability ranges from 0 to 1.
Discrete Probability Distribution includes the following:
A Random Variable is a numerical description of the outcome of an experiment.
Types of Random Variable:
Discrete Random Variable:
A Discrete Random Variable may assume either a finite number of values or an infinite sequence of values.
Continuous Random Variable:
A Continuous Random Variable may assume any numerical value in an interval or collection of intervals.
It is a weighted average of the Probability Distribution.
Multiplying the P of each outcome by the value of the outcome, and then summing the results.
Example:
Toss 2 coins, count the number of tails and compute expected value.
Solution:
Values
 Probability

0
 0.25

1
 0.50

2
 0.25

The weighted average squared deviation about the mean.
Example:
Toss 2 coins, count the number of tails and compute variance.
Solution:
Values
 Probability

0
 0.25

1
 0.50

2
 0.25

The Standard Deviation is the square root of the Variance.
Covariance is a statistic representation of the degree to which two variables vary together.
It shows the type and strength of the relationship between two variables.
Positive
 Indicates that the two variables will vary together in the same direction.
Negative
 Indicates that the two variables will vary in opposite directions.
Where,
Discrete Random Variable.
Outcome of .
Discrete Random Variable.
Outcome of .
Probability of occurrence of the outcome of and the outcome of .
It is used to find the probability that a given number of successes X occurs over a certain number of trialsn, each of which has the same probability p.
Success refers to the event of interest to us, such as:
 Getting a head when flipping a coin.
 Getting a question correct on a quiz.
Let x denote the number of successes occurring in the n trials.
Where,
 f(x) = The probability of x successes in n trials.
 p = The probability of success on any one trial.
Let X be a discrete random variable with the binomial distribution with parameters n and p.
The mean of binomial Distribution is given by:
Variance and Standard Deviation of Binomial Distribution arrow_upward
Let X be a discrete random variable with the binomial distribution with parameters n and p.
The variance and standard deviation of binomial Distribution is given by:
The Poisson Distribution is used to model the number of events occurring within a given time interval.
Properties of Poisson Experiment:
The probability of an occurrence is the same for any two intervals of equal length.
The expected value of occurrences in an interval is proportional to the length of this interval.
The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval.
The probability of two or more occurrences in a very small interval is close to 0.
Where,
is an average rate of value.
x is a poisson random variable.
e is the base of logarithm (e = 2.718).
The HyperGeometric Distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.
Where,
 A = Number of successes in the population.
 N – A = Number of failures in the population.
 X = Number of successes in the sample.
 n – X = Number of failures in the sample.