Continuous Probability Distribution

1. 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.

2. 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.

3. 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.

4. µ Normal Probability Distribution • PDF: • Shape: • Bell-Shaped and Symmetric • Mean, median and mode are equal

5. Normal Probability Distribution • Our notation for a random variable X that has mean m and variance s2 (and standard deviation s) is:

6. 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.

7. Finding za • za is the value of Z such that the area to the right is equal to a • Using symmetry, you can show that za =-z1-a. • Note that za is the 100*(1-a)th percentile of Z. • Example: z0.05 is the 95th percentile of Z.

8. Applications of the Normal Distribution • Any normal random variable X~N(m,s2) can be “standardized” into a standard normal random variable Z.

9. Percentiles of a Normal Distribution • Steps to finding the percentile of a normal distribution: • Find the percentile of the standard normal distribution (Z) which corresponds to the desired percentile. • Convert the standard normal percentile to the desired normal distribution with the following formula:

10. 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.

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

12. CLT (continued) • 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.

13. Standardizing for the CLT • Z formulae for the CLT include:

14. Normal Approximations • Binomial Distribution • Poisson Distribution