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 1 Flip 2 Flip 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.

Outcomes X Probability(H,H,H) 3(H,H,T) 2(H,T,H) 2(T,H,H) 2(H,T,T) 1(T,H,T) 1(T,T,H) 1(T,T,T) 0

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?

,,,2,1,0,)1(}{ nxppx

nxXP xnx

.)!(!

!

xnx

n

x

nwhere

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:

x

i

iXP1

}{

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

0 6 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?

• 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).

03

limlim}{ NN

nEP N

EN

The Normal Random Variable

• Bell shaped curve

-3 -2 -1 0 1 2 3

xexf

x2

2

1

2

1)(

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.)