# Goals ```Goals
Notes
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Construct probability distributions and graphs of discrete random
variables.
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Find the mean, variance, and standard deviation of discrete
probability distibutions.
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Finding expected values, which are the same as means.
Random variables are a type of variable similar to ones you may have
seen before. What makes them different is that a random phenomena
will result in an outcome that is a count or measurement.
Probability Distribution
Notes
Discrete probability distributions lists each possible value the random
variable can assume, together with its probability. It must satisfy:
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For any value x, 0 ≤ P(x) ≤ 1.
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Using all possible x, ΣP(x) = 1.
Ex: A personality trait test is administered to 150 employees. The
scores range from 1 (most passive) to 5 (most aggressive). The
results are shown below.
Score x
1
2
3
4
5
Frequency f
24
33
42
30
21
Probability Distribution
Notes
In other words, the probability distributions are frequency distribution
tables with:
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Each outcome is a single number. This replaces a class interval.
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Relative frequencies are probabilities.
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Relative frequency histograms are a type of graph of the
distribution.
Ex: Days of rain for a three day forecast:
Days of rain x
0
1
2
3
Probability P(x)
0.216
0.432
0.288
0.064
Ex: Decide whether the distribution is a probability distribution:
Notes
1.
x
5
P(x) 0.28
2.
x
P(x)
3.
x
P(x)
4.
x
5
P(x) 0.09
6
0.21
1
2
3
1
2
1
4
5
4
7
0.43
4
−1
1
2
3
4
1
16
5
8
1
4
1
16
6
0.36
8
0.15
7
0.49
8
0.06
The mean of a discrete random variable is given by
Notes
&micro; = Σx &middot; P(x)
This is from the weighted mean formula, using x as a value the
random variable can assume. This is also the same as the expected
value of a random variable.
Ex: The mean of the personality test is 2.94. What can you conclude?
Ex: The mean of the three day forecast is 1.2 What can you
conclude?
Ex: A company tracks the number of sales new employees make each
day during a 100-day period. The results are given below. Find the
mean. What can you conclude?
Sales per day x
Number of days f
0 1
16 19
2
15
3 4
21 9
5 6 7
10 8 2
The variance is given by the formula
σ 2 = Σ(x − &micro;)2 &middot; P(x)
The standard deviation is
√
p
σ = σ 2 = Σ(x − &micro;)2 &middot; P(x)
Ex: The variance and standard deviation for the personality test are
1.616 and 1.3 respectively. How do you interpret these numbers?
Ex: A table for the three day forecast is given. Fill in the blanks and
find the variance and standard deviation. Remember &micro; = 1.2
x
0
1
2
3
P(x) x − &micro; (x − &micro;)2
0.216
0.432
0.288
0.064
P(x) &middot; (x − &micro;)2
Notes
Expected Value
Notes
Expected value is the term to used to expect what would happen over
a large repetion of trials.
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It is the same as the mean,
Expected value = E (x) = &micro; = Σx &middot; P(x)
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Although probabilities are never negative, the expected value can
be negative.
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An expected value of 0 has a practical interpretation in games of
chance. What is it?
Ex At a raffle, 1500 tickets are sold at \$4 each. The prizes are \$750,
\$500, \$150, and \$100. You buy one ticket. Write a probability
distribution for the possible gains (or outcomes). What is your