Ch4: 4.3The Normal distribution 4.4The Exponential Distribution

Continuous pdfs, CDFs and Expectation Section In short continuous random variables are similar to discrete ones but with rules as follows: CDF and Expectation:

The Normal Distribution Section 4.3 The most important distribution of classical and applied statistics. Many experiments result in data that can be modeled using the normal distribution including but not limited to: height, weight, measurements of error in scientific experiments, reaction times in psychological experiments, test scores, … etc. Statistics such as means, proportions and variances estimated from samples also have a normal distribution given some conditions that we will learn about soon! Many of the discrete and continuous distributions can be approximated using the Normal under certain condition as well.

The Normal Distribution Section 4.3 1)Experiments of interest 2)Sample space 3)Random variable Many!! As long as the shape of the distribution is symmetrical and bell-shaped. Or assumed to be with the chance of observing the too large and too small values being very small

The Normal Distribution Section 4.3 4)Probability distribution We say that a random variable is normally distributed,, governed by two parameters the mean μ and the standard deviation σ, if the pdf of its distribution is, CDF Expectation

The Normal Distribution Section 4.3 The pdf for a mean = μ = 70 and Standard deviation = σ = 5 is, μ = 70 σ = 5

The Normal Distribution Section 4.3 Example: Suppose that the distribution of heights of adult men residing in the U.S. follows a normal distribution with mean = μ = 70in and standard deviation = σ = 2.5in. If we choose a man at random from the U.S. what is the chance that he will have a height equal to 65in?

The Normal Distribution Section 4.3 What is the chance that he will have a height less than 65in?

The Normal Distribution Section 4.3 What is the chance that he will have a height greater than or equal to 75in?

The Normal Distribution Section 4.3 What is the chance that he will have a height between 65in and 75in?

The Normal Distribution Section 4.3 To avoid this we use a standard normal distribution If we change the mean and standard deviation in the above problem we will obtain a new distribution. Still bell-shaped and symmetrical though will need us to find new integrals. Which can make life very tedious at times.

The Normal Distribution Section 4.3 The standard Normal To standardize we shift the normal distribution to force the mean to equal to zero and scale the variance and force it to equal to one! μ = 70 σ = 5 μ = 0 σ = 1

The Normal Distribution Section 4.3 The standard Normal To do so we define a new random variable Z that is a function of X, Z is said to have a standard normal distribution with mean = μ = 0 and standard deviation = σ = 1, pdf, A CDF,, as provided by Table A.3 pages

The Normal Distribution Section 4.3 The standard Normal Using the standard normal, find the probability Z < -1, Z > 1, 2.5 < Z < 3 -2 < Z <1 Z > -1.5 and Z < 2.5

The Normal Distribution Section 4.3 The standard Normal If we choose a man at random from the U.S. what is the chance that he will have a height equal to 65in? What is the chance that he will have a height less than 65in? What is the chance that he will have a height greater than or equal to 75in? What is the chance that he will have a height between 65in and 75in?

The Normal Distribution Section 4.3 Percentiles By definition the (100p)th percentile of a distribution is denoted by x(p) (η(p) in the book) and is the value of the random variable X such that the probability to the left that value is equal to p. In other words it is x(p) such that,

The Normal Distribution Section 4.3 Percentiles Find the 5 th percentile for the standard normal. Find the 97 th percentile for the standard normal Find the 50 th percentile for the standard normal. In the height example, find the 30 th percentile. In the height example, find the 99 th percentile

The Normal Distribution Section 4.3 z α = x(1-α) = equal to the (1-α) th percentile; we will get back to this when we start constructing confidence intervals and testing hypotheses. At that point I will emphasize the percentiles a bit more. Find the z 0.05

The Normal Distribution Section 4.3 Approximating the Binomial If, with, then we can use the normal distribution to approximate this distribution as follows, That is X is approximately normal with

The Normal Distribution Section 4.3 Example: Suppose that 25% of all licensed drivers in the state of Washington do not have insurance. If we intend to observe 50 drivers sampled at random from WA, what is the chance that we will observe 10 drivers or less that are uninsured (approximately and exactly)?