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Section Introduction to Random
Variables and
5.1
Probability Distributions
Random Variables
- measurements/counts obtains from a statistical
experiment
- Let x represent the quantitative result
- Ex: # of eggs in a robin’s nest or
- Ex: daily rainfall in inches
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Two Types of RVs
Discrete: things you count, usually whole numbers
Ex: # of students in a statistics class
Continuous: things you measure, usually fractions or
decimals
Ex: air pressure of a tire
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Probability Distribution
- it’s assigning probabilities to each measurement/count
- sum of the probabilities must be 1
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Ex 1 – Discrete RV
Dr. Mendoza developed a test to measure boredom
tolerance. He administered it to a group of 20,000 adults
between the ages of 25 and 35.
The possible scores were
0, 1, 2, 3, 4, 5, and 6, with
6 indicating the highest
tolerance for boredom.
a) Find the probabilities for
each score.
b) Are the scores mutually exclusive?
c) A company needs to hire someone with a score on the
boredom tolerance test of 5 or 6 to operate the fabric press
machine. Find the probability of someone scoring a 5 or 6.
d) Calculate the expected score (mean score).
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b. The graph of this distribution is simply a relative-frequency
histogram in which the height of the bar over a score
represents the probability of that score.
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- A probability distribution has a mean and standard
deviation
- The mean is called the “expected value”
- The standard deviation is thought of as “risk”
- The larger it is the greater the likelihood RV x is
different from the mean
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Expected Value for discrete RV
   x  P(x )
• Multiply each value times the probability of getting that
value.
• Add up the products.
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Standard Deviation for discrete RV
• Subtract each x from the mean.
• Square the difference.
• Multiply each squared difference by it’s corresponding
probability
• Add up all the products
• Square root the result
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Ex: 2
Are we influenced to buy a product by an ad we saw on
TV?
National Infomercial Marketing Association determined the
number of times buyers of a product had watched a TV
infomercial before purchasing the product. The results are
shown here:
Compute the mean and standard deviation of the
distribution.
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Linear Combinations
Recall:
**Multiplying/adding something to data values, the mean
changes by the same amount
**Only multiplying something to data values changes the st.
dev.
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Ex: 3
Let x1 and x2 be independent RVs with means 1 = 75 and
2 = 50, and st.dev. 1 = 16 and 2 = 9.
a) Let L = 3 + 2x1. Compute the mean, variance, and st.dev
of L.
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cont’d
b) Let W = x1 + x2. Find the mean, variance, and st.dev of
W.
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c) Let W = x1 – x2. Find the mean, variance, and st.dev. of
W.
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d) Let W = 3x1 – 2x2. Find the mean, variance, and st. dev.
of W.
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Linear combination formulas
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Ex: 4
A tool of cryptanalysis (science of
code breaking) is to use relative
frequencies of occurrence of letters
to break codes. In addition to
cryptanalysis, creation of word
games also uses the technique.
Oxford Dictionaries publishes
dictionaries of English vocabulary.
They did an analysis of letter
frequencies in words listed in the
main entries of the Concise Oxford
Dictionary. Suppose someone took
a random sample of 1000 words
occurring in crossword puzzles.
Find the P(letter will be a vowel):
Lette
r
Freq. Prob. Lette
r
Freq. Prob.
A
85
N
66
B
21
O
72
C
45
P
32
D
34
Q
2
E
112
R
76
F
18
S
57
G
25
T
69
H
30
U
36
I
75
V
10
J
2
W
13
K
11
X
3
L
55
Y
18
M
30
Z
3
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Ex: 5
At a carnival, you pay $2.00 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 $1.00 for every 1 that
appears on top.
a) What is the random variable?
b) What is the sample space (all outcomes)?
c) Calculate the expected earning if you play this game. Is
the expected earning less than, equal to, more than the
cost of the game?
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Section 5.1 Assignment:
pg.205: #1, 3, 7, 8, 11, 13, 15, 17, 19,
21(omit part d)
You must show your work!
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