1. Probability Distributions, Discrete Random Variables
Objectives: (Chapter 5, DeCoursey)
- To define a probability function, cumulative probability, probability distribution function and cumulative distribution functions.
- To define expectation and variance of a random variable.
- To determine probabilities by using Binomial Distribution.

2. Probability Distributions, Discrete Random Variables
Probability function:
Consider a random discrete variable, X, which can only take on certain values within an interval of interest, such as x0, x1, x2, …xk,
The corresponding probabilities are p(x0), p(x1), p(x2),… p(xk). Then p(xi) (i=0,1,…k) or Pr[X=xi] is called a probability function.

3. Probability Distributions, Discrete Random Variables
Probability Function:
The graph or table listing the various values of p(xi) is the probability distribution function for the variable x. (e.g. tossing 2 fair coins)
X, no. of heads
Probability, p(x)
1/4 1
2/4 2
Total
p(xi)
0.75
0.5
0.25 1 2 X

4. Probability Distributions, Discrete Random Variables
Cumulative Probabilities: X: random variable, x: upper limit
Sometimes it is useful to know the probability that a
variable will take on a value below some upper limit.
Cumulative probability can be used.

5. Probability Distributions, Discrete Random Variables
Cumulative Distribution Function: The functional relationship between the cumulative probability and the upper limit, x.
Pr[X≤x]
1.0
X, no. of heads
Probability, p(x)
1/4 1
2/4 2
Total
0.75
0.5
0.25 1 2

6. Probability Distributions, Discrete Random Variables
Expectation of a Random Variable is the mean value of the distribution of the random variable.
It is an arithmetic mean that we can expect to closely approximate the mean result from a very long series of trials, if a particular probability function is followed.
The expectation of a random variable X is denoted by E(X) or μx or μ.

7. Probability Distributions, Discrete Random Variables
Variance of a Discrete Random Variable:
Standard Deviation:

8. Probability Distributions, Discrete Random Variables
Binomial Distribution: A special type of probability distribution. It is applicable to situations where:
There are only two possible outcomes for a given trial. e.g. heads or tails, good or bad, success or failure.
The outcome is determined completely by chance.
All trails have the same probability for a particular outcome in a single trial. i.e. the probability in a subsequent trial is independent of the outcome of a previous trial. Let this constant probability for a single trial be p.
Apply to quality control.

9. Probability Distributions, Discrete Random Variables
Binomial Distribution:
Let p = probability of “success”
q = probability of “failure” = 1-p
n = number of trials
r = number of “success” in “n” trials
Then the probability of r successes for n trials is given by the following general formula:

10. Probability Distributions, Discrete Random Variables
Binomial Distribution:
Expectation:
Standard deviation: