mba512 probability distributions

Random Variables

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes • How to summarize a random variable?

– The mean of a probability distribution is called the expected value

– The standard deviation

Random Variables

Discrete Random Variable

ContinuousRandom Variable

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes

Random Variables

Discrete Random Variable

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes

Binomial distribution (n = 100, p = 0.3)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

X

p(X

)

Counting the expected number of

successes by specifying the probability of

success

Binomial DistributionPoisson distribution (µ = 5)

0.00

0.020.04

0.06

0.080.10

0.12

0.14

0.160.18

0.20

0 2 4 6 8 10 12 14 16 18 20

X

p(X

)

Counting the expected number of arrivals by

specifying the approximate arrival

rate

Poisson Distribution

Random Variables

Discrete Random Variable

ContinuousRandom Variable

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes

Uniform Distribution

Describes measurements that are equally likely

Exponential Distribution

Measurements of inter-arrival times

for the arrivals described by the Poisson process

The Normal Distribution

• ‘Bell Shaped’ & Symmetric• Mean, Median and Mode are equal • Location is characterized by the mean, μ• Spread is characterized by the standard deviation, σ

The random variable has an infinite theoretical range: - to +

Mean = Median = Mode

p(X)

μ

σ

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

2μ)(X

2

1

e2π

1p(X)

The Normal Distribution

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

The Normal Distribution

The next step would be to develop a probability distribution for your sample data (which would require that you calculate probabilities of all outcome categories)…

The convenience of named probability distributions is that determining probabilities can be accomplished with a formula or table (so you need to determine which probability distribution best describes your data)…

Random Variables

Discrete Random Variable

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes

Binomial distribution (n = 100, p = 0.3)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

X

p(X

)

Counting the expected number of

successes by specifying the probability of

success

Binomial Distribution

=BINOMDIST(k, n, p, 0 or 1)

Poisson distribution (µ = 5)

0.00

0.020.04

0.06

0.080.10

0.12

0.14

0.160.18

0.20

0 2 4 6 8 10 12 14 16 18 20

X

p(X

)

Counting the expected number of arrivals by

specifying the approximate arrival

rate

Poisson Distribution

=POISSON(k, l, 0 or 1)

Random Variables

Discrete Random Variable

ContinuousRandom Variable

Probability Distributions

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

A probability distribution is a table of all possible outcomes for a variable and the associated probability.

Random variables assign numerical values to the outcomes

Uniform Distribution

Describes measurements that are equally likely

=1/(b-a)

Exponential Distribution

Measurements of inter-arrival times

for the arrivals described by the Poisson process

=EXPONDIST(k, 1/l, 1)

The Normal Distribution

The Normal Distribution

m-s s-2s 2s-3s 3s

The Normal Distribution

m-s s-2s 2s-3s 3s

The Normal Distribution

m-s s-2s 2s-3s 3s

The Normal Distribution

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

To find P(m<a) • =NORMDIST(a, , m s, 1)

To find a: P(m<a) = p • =NORMINV(p, , ,1)m s

p(X)

μ

σ

2μ)(X

2

1

e2π

1p(X)

The Standard Normal Distribution

• ‘Bell Shaped’ & Symmetric• Mean, Median and Mode are equal • The mean, μ, is always 0• The standard deviation, σ , is always 1

The std normal distr. tells how far from the mean the observation lies…

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

Z

p(Z)

0

1

2

Z2

e2π

1p(Z)

To find P(z<a) • =NORMSDIST(

a)To find a: P(z<a) = p • =NORMSINV(

p)

The Standard Normal Distribution

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

Z

p(Z)

0

1

2

Z2

e2π

1p(Z)

To find P(z<a) • =NORMSDIST(

a)To find a: P(z<a) = p • =NORMSINV(

p)

…allows us to convert any set of measurements to a common scale

…standardizingTo standardize• =STANDARDIZE(a, , m

s)

• ‘Bell Shaped’ & Symmetric• Mean, Median and Mode are equal • The mean, μ, is always 0• The standard deviation, σ , is always 1

The std normal distr. tells how far from the mean the observation lies…

The Standard Normal Distribution

0-1 1-2 2-3 3

Sure would be nice if my data were Normal…

Sampling Distribution of the Sample Mean

• A sampling distribution of the sample mean is a collection of sample means (of a given sample size)

How should we describe the sampling distribution?

– The mean of a sampling distribution is the expected value, E(X)

– A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean

n

σσ

X

Sampling Distribution of the Sample Mean

• A sampling distribution of the sample mean is a collection of sample means (of a given sample size)

• GOAL OF SAMPLING: for the mean of the sampling distribution to equal the true population mean

…an unbiased sample

E(X) =

Central Limit Theorem

• If samples of a given size are selected from any population, the sampling distribution of the sample mean will be normally distr.

Population Distribution

μ

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

Central Limit Theorem

• If samples of a given size are selected from any population, the sampling distribution of the sample mean will be normally distr.

• As n gets larger and larger, this approximation gets better Sampling Distribution

(becomes normal as n increases)Larger sample sizeSmaller

sample size

xμ

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

Sampling from non-Normal Distributions

• For most distributions, n > 30 will give a sampling distribution that is nearly normal

• For fairly symmetric distributions, n > 15 will give a sampling distribution that is nearly normal

• For normal population distributions, the sampling distribution of the mean is always normally distributed

Some material from ppt slides for Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.

Sampling Distribution of the Sample Proportion

Proportions are also normally distributed…

• Expected value for the proportion: mp

= p

• Standard error for the proportion:n

)(1σp