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.)
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