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 T H T H T H T H T H T H T Flip 1Flip 2Flip 3 (H,H,H) (H,H,T) (H,T,H) (H,T,T) (T,T,T) (T,H,H) (T,H,T) (T,T,H)

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 (f x ), 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.

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? Therefore, P{6 < W < 9} = Continuous RV’s - Example 06912

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?

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 Continuous RV’s - Example (cont).

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 (f x ), 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.)