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

Normal Probability Distribution  Our notation for a random variable X that has mean  and variance   (and standard deviation  ) is:

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.

Finding z   z  is the value of Z such that the area to the right is equal to   Using symmetry, you can show that z  =-z 1-   Note that z  is the 100*(1-  ) th percentile of Z.  Example: z  is the 95 th percentile of Z.

Applications of the Normal Distribution  Any normal random variable X~N( ,  2 ) can be “standardized” into a standard normal random variable 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:

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  and variance  2 >0, then 1. the sample mean will follow a normal distribution with mean  and variance  2 /n,

CLT (continued) 2. the sum of the data will follow a normal distribution with mean n  and variance n  2.  The CLT can be used with any sample size if the underlying data follows a normal distribution.

Standardizing for the CLT  Z formulae for the CLT include:

Normal Approximations  Binomial Distribution  Poisson Distribution