Class 3 Binomial Random Variables Continuous Random Variables

Class 3 Binomial Random Variables Continuous Random Variables Standard Normal Distributions

Random Variables • Recall that a probability distribution is a list of the values and probabilities that a random variable assumes. • These values can be thought of as the values in a population, and the probabilities as the proportion of the population that a specific value makes up. • Random variables can be classified as being discrete or continuous. Continuous random variables assume values along a continuum.

Binomial Random Variables • Certain random variables (populations) arise frequently in studying probabilistic situations. These random variables have been given special names. • The random variable that we studied that represented the number of heads observed in three flips of a coin was actually an example of a binomial random variable.

A Detailed Look at the Coin Flips H H T (H, H, H) T (H, H, T) H (H, T, H) T H (H, T, T) (T, H, H) T H (T, H, T) (T, T, H) T (T, T, T) T H T Flip 1 H Flip 2 Flip 3

Binomial RV (cont. ) • Characteristics of experiments that lead to binomial random variables • The experiment consists of n identical and independent trials. • Each trial results in one of only two possible outcomes, say success or failure. • If X = the number of successes in n trials, then X is said to have a binomial distribution.

Binomial RV (cont. ) • Examples • It rains one out of every 4 days in the summer in Ohio. We select 5 days at random. Let X = the number of days it rains out of 5. • 20% of all bolts produced by a machine are defective. We select 30 bolts. Let X = the number of defective bolts. • Flip a coin three times. Let X = the number of heads observed.

Our Example Let p = P{head}.

Binomial RV (cont. ) • Let p = P{success} at each of n trials. Then • = np, 2 = np(1 -p) • Do these formulae work for our coin flip example?

Using EXCEL to compute Binomial Probabilities • Select the Function Wizard (fx), statistical/binomdist • The syntax for this function is binomdist(x, n, p, true or false). • If the fourth argument is false, it will return P{X=x} for a binomial with parameters n and p. • If the fourth argument is true, it will return the cumulative distribution to x:

Summary on Discrete RV’s • There are many different types of discrete random variables • • Binomial Uniform Poisson Hypergeometric • A probability distribution serves as a model of what the population looks like.

Continuous Random Variables • Instead of a probability distribution, a density function describes the density of the values in the population. • The area under the density function is the probability of an event.

Continuous RV’s - Example • The amount of gasoline in my gas tank, W, is between 0 and 12 gallons. Suppose every value has the same chance of occurring. What is p{0 < W < 12}? What does this imply about the function? 0 6 9 • Therefore, P{6 < W < 9} = 12

Continuous RV’s - Example (cont). • Can you describe this population in words? • What is the P{W = 6}? • What would the density function look like (generally) for a person who tended to keep their tank full?

Continuous RV’s - Example (cont). • An event has probability 0 if it happens a finite number of times in an infinite number of trials. • Recall the idea of relative frequency. If an event E only happens, say, 3 times in an infinite number of trials, then

The Normal Random Variable • Bell shaped curve

Normal RV’s (cont. ) • It turns out that the two parameters in this function, and , have the natural interpretations: if X has a normal distribution, then E(X) = , and Var(X) = 2. • The function is completely specified by and , thus a normal distribution is completely specified by its mean and variance.

Normal RV’s (cont. ) • The area (probability) under this bell shaped curve is difficult to determine. As a result, tables of areas have been determined for the case = 0 and = 1 (called Z, the standard normal random variable). • The probability computation for any other normal distribution ( 0 or 1) has to be converted to one about Z. • The can also be done in EXCEL.

Computing Standard Normal Probabilities Therefore, P{Z<1. 14} =. 8729

Computing Normal Probs. • P{1. 14 < Z} = • P{Z < -1. 14} = • P{-1. 14 < Z < 0} = • P{Z < 1. 14} =

Computing Standard Normal Probabilities in EXCEL • Select Function Wizard (fx), statistical/normsdist • The function normsdist takes an argument, z, and returns the area under the standard normal distribution to the left of z • The function normsinv takes an area (probability) and returns the value that cuts off that area to the left. (This is the inverse of normdist. )