Transcript Class 3 Handout

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,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
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}.
Outcomes
(H,H,H)
(H,H,T)
(H,T,H)
(T,H,H)
(H,T,T)
(T,H,T)
(T,T,H)
(T,T,T)
X
3
2
2
2
1
1
1
0
Probability
Binomial RV (cont.)
• Let p = P{success} at each of n trials. Then
n x
P{ X  x}    p (1  p)n  x , x  0,1, 2,, n,
 x
n
where   
 x
n!
.
x!(n  x )!
•  = 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:
x
 P{ X  i}
i 1
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
nE
3
P{E}  lim N 
 lim N   0
N
N
The Normal Random Variable
• Bell shaped curve
-3
-2
f ( x) 
-1
1
e
 2
0
1
1 x   
 

2  
2
3
2
  x
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.)