Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Distributions Probability Distributions BUS-121 November, 2013 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Probability Distributions This short PPT is intended to help students with their understanding and learning of probability distributions used in this course BUS-121. This presentation cannot be used for understanding without reference to the full presentations, the course textbooks or tutors. It is intended as a useful summary of the main characteristics of each selected distribution but without explanation or attempt at full description. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Probability Distributions A probability distribution is a table, formula, or graph that describes the values of a random variable and the probability associated with these values. Since we’re describing a random variable (which can be discrete or continuous) we have two types of probability distributions: – Discrete Probability Distribution, (Chapter 7) and – Continuous Probability Distribution (Chapter 8) Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Distributions of Each Type Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Distributions of Each Type Discrete Continuous Binomial Normal Poisson Uniform Exponential Students’ t Chi Squared F These are the distributions to learn for this course Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Binomial Distribution Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Binomial Distribution Statisticians have developed general formulas for the mean, variance, and standard deviation of a binomial random variable. The Binomial Distribution can be approximated by the Normal Distribution for large values of n Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Poisson Distribution The probability that a Poisson random variable assumes a value of x is given by: and e is the natural logarithm base. The mean and variance are the same value Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 The Normal Distribution Important things to note: The normal distribution is fully defined by two parameters: its standard deviation and mean The normal distribution is bell shaped and symmetrical about the mean Unlike the range of the uniform distribution (a ≤ x ≤ b) Normal distributions range from minus infinity to plus infinity Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Standard Normal Distribution A normal distribution whose mean is zero and standard deviation is one is called the standard normal distribution. As we shall see shortly, any normal distribution can be converted to a standard normal distribution with simple algebra. This makes calculations much easier. 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Uniform Distribution Consider the uniform probability distribution (sometimes called the rectangular probability distribution). It is described by the function: f(x) a b x area = width x height = (b – a) x = 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Exponential Distribution Another important continuous distribution is the exponential distribution which has this probability density function: Note that x ≥ 0. Time (for example) is a non-negative quantity; the exponential distribution is often used for time related phenomena such as the length of time between phone calls or between parts arriving at an assembly station. For the exponential random variable: The mean and standard deviation are the same value Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Student t Distribution… Here the letter t is used to represent the random variable, hence the name. The density function for the Student t distribution is as follows… ν (nu) is called the degrees of freedom, and Γ (Gamma function) is Γ(k)=(k-1)(k-2)…(2)(1) 8.11 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Student t Distribution… Much like the standard normal distribution, the Student t distribution is “mound” shaped and symmetrical about its mean of zero: The mean and variance of a Student t random variable are E(t) = 0 and V(t) = for ν > 2. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Chi-Squared Distribution… The chi-squared density function is given by: As before, the parameter is the number of degrees of freedom. Figure 8.27 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 F Distribution… The F density function is given by: F > 0. Two parameters define this distribution, and like we’ve already seen these are again degrees of freedom. is the “numerator” degrees of freedom and is the “denominator” degrees of freedom. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 F Distribution… The mean and variance of an F random variable are given by: and The F distribution is similar to the distribution in that its starts at zero (is non-negative) and is not symmetrical. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Central Limit Theorem (Chapter 9) The sampling distribution of the mean of a random sample drawn from any population is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Keller: Stats for Mgmt & Econ, 7th Ed November 9, 2013 Central Limit Theorem (Chapter 9) If the population is normal, then X is normally distributed for all values of n. If the population is non-normal, then X is approximately normal only for larger values of n. In most practical situations, a sample size of 30 may be sufficiently large to allow us to use the normal distribution as an approximation for the sampling distribution of X. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.