Transcript Binomial distribution - Pennsylvania State University

Binomial Distribution
And general
discrete probability distributions...
Random Variable
• A random variable assigns a number to a
chance outcome or chance event.
• The definition of the random variable is
denoted by uppercase letters at the end of
alphabet, such as W, X, Y, Z.
• The possible values of the random variable
are denoted by corresponding lowercase
letters w, x, y, z.
Examples:
Discrete random variables
• W = number of beers randomly selected
student drank last night, w = 0, 1, 2, …
• X = number of aspirin randomly selected
student took this morning, x = 0, 1, 2, ...
• Y = number of children in a family with
children, y = 1, 2, 3, ...
Discrete Probability Distribution
• A discrete probability distribution
specifies:
– the possible values of the random variable, and
– the probability that each outcome will occur
Example:
Discrete probability distribution
• Let X = number of natural brothers PSU
students have. X = 0, 1, 2, ...
• P(X=0) = 0.41
• P(X=1) = 0.45
• P(X=2) = 0.11
• P(X=3) = 0.03
Example: Graphically …
0.45
0.40
Probability
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
X, number of brothers
3
Example:
Discrete probability distribution
• Let X = the number selected when a student
picks a number between 0 and 9.
• If students pick number randomly, then the
probability of picking any number is 0.10.
• That is, P(X = 0) = … = P(X = 9) = 0.10
Example:
Binomial random variable
•
•
•
•
Student told 3 statements -- 2 true, 1 false.
Student tries to identify false statement.
Student does this 3 different times.
Let X = the number of times student
correctly identifies the false statement.
Then, X = 0, 1, 2 or 3.
Binomial Random Variable
A special kind of discrete random variable having
the following four characteristics:
• n identical “trials”: student tries to guess 3 times
• 2 possible outcomes denoted “success” or “failure”:
student picks either false statement or true statement.
• independent trials: a student’s success or failure on one try
doesn’t affect success or failure on another try
• p = P(“success”) is same for each trial: if just guessing, a
student has probability of 1/3 of picking false statement.
Is X binomial?
Probability student smokes pot regularly is
0.25.
College administrator surveys students until
finds one who smokes pot.
Let X = number of students surveyed.
Is X binomial?
Unknown to quality control inspector, crate
of 50 light bulbs contain 3 defective bulbs.
QC inspector randomly selects 5 bulbs
“without replacement”.
Let X = number of defective bulbs in
inspector’s sample.
Is X binomial?
Unknown to us, the probability an American
thinks Clinton should have been removed
from office is 0.29.
Gallup poll surveys 960 Americans.
Let X = number of Americans in sample
who think Clinton should have been
removed from office.
Is X binomial?
Students pick one number between 0 and 9.
Let X = number of students who pick the
number “7”
Example: Binomial r.v.
Let 3 students pick. Let Y = #7 and N = not #7
Outcome
X
P(X=x)
NNN
0
1(0.1)0(0.9)3
YNN
NYN
NNY
YYN
YNY
NYY
YYY
1
3(0.1)1(0.9)2
2
3(0.1)2(0.9)1
3
1(0.1)3(0.9)0
Binomial Probability Distribution
P(X = x)
= (# of ways x occurs) × px × (1-p)n-x
= n!/[x!(n-x)!] × px × (1-p)n-x
Where “n-factorial” is defined as
n!= n (n-1) (n-2) … 1
and
0! = 1
Examples: n!
5! = 5 × 4 × 3 × 2 × 1 = 120
4! = 4 × 3 × 2 × 1 = 24
3! = 3 × 2 × 1 = 6
2! = 2 × 1 = 2
1! = 1
Example: Binomial Formula
P(X = x) = n!/[x!(n-x)!] × px × (1-p)n-x
Guessing game. Let n = 3 and p = 0.33. Then:
P(X = 0) = 3!/[0!(3-0)!] × 0.330 × (0.67)3-0
= 6/(1×6) × 1 × 0.673 = 0.30
P(X = 1) = 3!/[1!(3-1)!] × 0.331 × (0.67)3-1
= 3 × 0.33 × 0.672 = 0.44
Example (continued)
P(X = 2) = 3!/[2!(3-2)!] × 0.332 × (0.67)3-2
= 3 × 0.1089 × 0.67 = 0.22
P(X = 3) = 3!/[3!(3-3)!] × 0.333 × (0.67)3-3
= 1 × 0.037 × 1 = 0.04
Note: 0.30 + 0.44 + 0.22 + 0.04 = 1 
Using binomial probabilities to
draw a conclusion
If students did just randomly guess which
statement was false, we’d expect the
random variable X to follow a binomial
distribution?
Can we conclude that students did not guess
the false statements randomly?
Using binomial probabilities to
draw a conclusion
If students do indeed pick a number
between 0 and 9 randomly, how likely is it
that we would observe the sample we did?
Can we conclude that students do not pick
numbers randomly?
Using binomial probabilities to
draw a conclusion
Could the space shuttle Challenger disaster of
January 28, 1986 have been better predicted?
And therefore prevented?
Moral
• Probability calculations are used daily to
draw conclusions and make important
decisions.
• Calculated probabilities are accurate only if
the assumptions made are indeed correct.
• Always check to see if your assumptions
are reasonable.