Continuous Probability Distribution

A continuous random variables (RV) has infinitely many possible outcomes Probability is conveyed for a range of values,

not for individual values. Example: Satellite falling from orbit.

Uniform Probability Distribution All outcomes are equally likely. Shape of distribution is a rectangle.

Probability Density Function (PDF)

A PDF is used to convey probability for a continuous random variable. Area under the PDF indicates probability Total area under the PDF is 1 The PDF must be non-negative for all values The probability of an observation falling

between a and b is equal to the area under the PDF between a and b.

Normal (Gaussian) Distribution

Many real world applications utilize the normal distribution. Naturally occurs in test scores, experimental

errors, measures of sizes in populations, etc. Data that is summed or averaged can be

shown to follow a distribution.

Normal Probability Distribution

PDF:

Shape: Bell-Shaped and Symmetric Mean, median and mode are equal

µ

2

2

1

2

1)(

x

exf

Normal Probability Distribution

Our notation for a random variable X that has mean m and variance s2 (and standard deviation s) is:

),(N~ 2X

Standard Normal Distribution

A normal random variable with mean 0 and standard deviation 1 is denoted by Z and called a standard normal random variable.

Probabilities for Z are found using a standard normal probability table like A-2 in your book.

)1,0(N~Z

Finding za

z a is the value of Z such that the area to the right is equal to a

Using symmetry, you can show that z a =-z1- .a

Note that za is the 100*(1-a)th percentile of Z. Example: z0.05 is the 95th percentile of Z.

)( zZP

Applications of the Normal Distribution

Any normal random variable X~N(m,s2) can be “standardized” into a standard normal random variable Z.

x

Z

Percentiles of a Normal Distribution

Steps to finding the percentile of a normal distribution:

1. Find the percentile of the standard normal distribution (Z) which corresponds to the desired percentile.

2. Convert the standard normal percentile to the desired normal distribution with the following formula:

zX

Sampling Distributions

Recall that a statistic is random in value … it changes from sample to sample.

The probability distribution of a statistics is called a sampling distribution.

The sampling distribution can be very useful for evaluating the reliability of inference based on the statistic.

Central Limit Theorem (CLT)

If a random sample of sufficient size (n≥30) is taken from a population with mean m and variance s2>0, then

1. the sample mean will follow a normal distribution with mean m and variance s2/n,

nx2

,N~

CLT (continued)

2. the sum of the data will follow a normal distribution with mean nm and variance ns2.

The CLT can be used with any sample size if the underlying data follows a normal distribution.

2,N~ nnx

Standardizing for the CLT

Z formulae for the CLT include:

n

xZ

n

nxZ

Normal Approximations

Binomial Distribution

Poisson Distribution

),(),( npqnpNpnB

),()( NP