Download Lesson 6.2 Discrete Probability Distribution: Standard Deviation and

Lesson 6.2 Discrete Probability Distribution: Standard
Deviation and Mean Notes
Statistics
Page 1 of 4
Example 1:
One of the elementary tools of cryptanalysis (the science of code breaking) is to use
relative frequencies of occurrence of different letters in the alphabet to break standard
English alphabet codes. Large samples of plain text such as newspaper stories
generally yield about the same relative frequency for letters. A sample of 1000 letters
long yielded the information in the table below.
Letter Frequency Probability
A
73
B
9
C
Letter Frequency Probability
N
78
0.078
0.009
O
74
30
0.030
P
27
0.027
D
44
0.044
Q
3
0.003
E
130
R
77
0.077
F
28
0.028
S
63
0.063
G
16
0.016
T
93
0.093
H
35
0.035
U
27
I
74
V
13
0.013
J
2
0.002
W
16
0.016
K
3
0.003
X
5
0.005
L
35
0.035
Y
19
0.019
M
25
0.025
Z
1
0.001
a. Use the relative frequencies to compute the omitted probabilities in the table.
b. Do the probabilities of all the individual letters add up to 1?
c. If a letter is selected at random from a newspaper story, what is the probability
that the letter will be a vowel?
Lesson 6.2 Discrete Probability Distribution: Standard
Deviation and Mean Notes
Statistics
Page 2 of 4
Mean and Standard Deviation of a discrete population probability distribution
 Mean
o µ=∑xP(x); µ is called the expected value
o The mean is called the expected value
 Standard deviation
o σ = √∑(x - µ)2P(x)
where x is the random variable, P(x) is the probability of that variable and
the sum ∑ is taken for all the values of the random variable.
Example 2: Are we influenced to buy a product by an ad we saw on TV? The
National Infomercial Marketing Association determined the number of times buyers
of a product watched a TV infomercial before purchasing the product. The results
are shown here. Calculate the mean and the standard deviation.
Number of times buyers saw the infomercial 1
2
3
4
5
Percentage of buyers
27% 31% 18% 9% 15%
Example 3: At a carnival, you pay $2 to play a coin-flipping game with three fair
coins. On each coin, one side has the number 0 and the other side has the number 1.
You flip the three coins at one time, and you win $1 for every 1 that appears on top.
Are your expected earnings equal to the cost of play?
a. What is the sample space for the values of this random variable?
Lesson 6.2 Discrete Probability Distribution: Standard
Deviation and Mean Notes
Statistics
Page 3 of 4
b. There are eight equally likely outcomes for throwing three coins. List the
outcomes.
c. Make of frequency table including the probability and the product of the
probability and the number of 1’s.
d. Find the expected value of return.
Example 4: A quiz with possible scores 1, 2, 3, and 4 was given to a class of 25
students. The score distribution is listed in the table.
a. Part of the score distribution and corresponding probability distribution table
is given below. Complete the table.
Score (x)
Number of Students
Probability
1
0.2
2
6
3
10
4
0.16
b. Do the probabilities add up to 1?
c. If a student is selected from this class at random, what is the probability that
his or her score is below 3?
d. Find the mean and the standard deviation.
Lesson 6.2 Discrete Probability Distribution: Standard
Deviation and Mean Notes
Statistics
Page 4 of 4
Example 5: At a carnival, you pay $1 to play a coin-flipping game with two fair coins.
If you get Heads on both coins, then you will win $3. Are your expected earnings
equal to the cost of play?
a. What is the random variable of interest in this case?
b. What is the sample space for this random variable?
c. There are four equally likely outcomes for throwing two coins. What are they?
d. Complete the following table regarding the earning.
Earning, x
Frequency
P(x)
0
3
3
0.25
xP(x)
e. Find the expected earnings. Is that more than, equal to, or less than the cost to
play? What does that mean in the long run?
Assignment: p. 210 #6, 8, 9, 14