Probability Distribution

The probability distribution for a random variable is an assignment of probability to each of the possible values for the variable

Probability Distributions

Discrete Probability Distributions 1. The Uniform Distribution; 2. The Binomial Distribution; 3. The Hypergeometric Distribution; and 4. The Poisson Distribution

Continuous Probability Distributions 1. The Uniform Distribution; and 2. The Normal Distribution

Mean of a Discrete Probability Distribution

The mean of a discrete probability distribution is also called the expected value of the discrete random variable.

where, E(X) = expected value of XX = values of the random variableP(X) = probability of each value of

X

E X XP X( ) ( )

Standard Deviation of a Discrete Probability Distribution

Standard Deviation of a discrete random variable

where, X = values of the random variable

E(X) = expected value of X P(X) = probability of each value

of X

x SQRT x E x P x ( ) ( )2

The Binomial or Bernoulli Process

The experiment consists of n identical trials;Each trial results in one of two outcome,

success or failure;The probability of success on a single trial is

equal to p, and remains the same from trial to trial. The probability of failure is ( 1 - p );

The trials are independent; andThe experimenter is interested in X, the

number of successes observed during the n trials.

Example: Taste Test; Coke Vs. PepsiSample size (n) = 3, P(Coke is preferred) = 0.20

Simple ConsumerEvent One Two Three P(ei)_ ___

e1 C C C (0.20)3 = 0.008

e2 C C P (0.20)2(0.80) = 0.032

e3 C P C (0.20)2(0.80) = 0.032

e4 C P P (0.20)(0.80)2 = 0.128

e5 P C C (0.20)2(0.80) = 0.032

e6 P C P (0.20)(0.80)2 = 0.128

e7 P P C (0.20)(0.80)2 = 0.128

e8 P P P (0.80)3 = 0.512

1.000

Probability Distribution

Coke Vs. Pepsi_X_ ___ei___ _P(X)_

0 e8 0.512

1 e4,e6,e7 0.384

2 e2,e3,e5 0.096

3 e1 0.008

1.000