Transcript random variables

RANDOM VARIABLES
• Random variables
• Probability distribution
• Random number generation
– Expected value
– Variance
– Probability distributions
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RANDOM VARIABLES
• Random variable:
– A variable whose numerical value is determined by the
outcome of a random experiment
• Discrete random variable
– A discrete random variable has a countable number of
possible values.
– Example
• Number of heads in an experiment with 10 coins
• If X denotes the number of heads in an experiment
with 10 coins, then X can take a a value of 0, 1, 2,
…, 10
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RANDOM VARIABLES
– Other examples of discrete random variable: number of
defective items in a production batch of 100, number of
customers arriving in a bank in every 15 minute,
number of calls received in an hour, etc.
• Continuous random variable
– A continuous random variable can assume an
uncountable number of values.
– Examples
• The time between two customers arriving in a bank,
the time required by a teller to serve a customer,
etc.
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DISCRETE PROBABILITY DSTRIBUTION
• Discrete probability distribution
– A table, formula, or graph that lists all possible events and
probabilities a discrete random variable can assume
– An example is shown below:
Discrete Probability Distribution
Probability
0.75
0.5
0.25
0
HH
HT
Event
TT
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CONTINUOUS PROBABILITY DSTRIBUTION
• Continuous probability distribution
– Similar to discrete probability distribution
– Since there are uncountable number of events, all the
events cannot be specified
– Probability that a continuous random variable will assume
a particular value is zero!!
– However, the probability that the continuous random
variable will assume a value within a certain specified
range, is not necessarily zero
– A continuous probability distribution gives probability
values for a range of values that the continuous random
variable may assume
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f(x)
CONTINUOUS PROBABILITY DSTRIBUTION
z
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f(x)
CONTINUOUS PROBABILITY DSTRIBUTION
z
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REVISIT SIMPLE RANDOM SAMPLING
• In Chapter 5, a simple random sample of 10 families is
chosen from a group of 40 families.
– 40 Random numbers are generated
– Each random number is between 0 and 1 (not
including 1)
– Excel RAND() function is used to generate each
random number.
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REVISIT SIMPLE RANDOM SAMPLING
– What is the average of the random numbers
generated?
– What is the variance of the random numbers
generated?
– What is the standard deviation of the random numbers
generated?
E
9 Average
10 Variance
11 Standard deviation
F
0.5155
0.0856
0.292635444
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REVISIT SIMPLE RANDOM SAMPLING
– Plot a histogram with all the random numbers, and
comment on the distribution of the random numbers.
Frequency
Histogram
8
7
6
5
4
3
2
1
0
0.125 0.25 0.375
0.5
0.625 0.75 0.875
Random Numbers
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RANDOM NUMBER GENERATION
• Most software can generate discrete and continuous
random numbers (these random numbers are more
precisely called pseudo random numbers) with a wide
variety of distributions
• Inputs specified for generation of random numbers:
– Distribution
– Average
– Variance/standard deviation
– Minimum number, mode, maximum number, etc.
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RANDOM NUMBER GENERATION
• Next 4 slides
– show histograms of random numbers generated and
corresponding input specification.
– observe that the actual distribution are similar to but
not exactly the same as the distribution desired, such
imperfections are expected
– methods/commands used to generate random
numbers will not be discussed in this course
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: uniform distribution,
min = 500 and max = 800
Frequency
Uniform Distribution
25
20
15
10
5
0
Random Numbers
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: triangular distribution,
min = 3.2, mode = 4.2, and max = 5.2
Frequency
Triangular Distribution
120
100
80
60
40
20
0
3.2 3.4 3.6 3.8
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4.2 4.4 4.6 4.8
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5.2
Radom Numbers
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: normal distribution,
mean = 650 and standard deviation = 100
30
25
20
Random Numbers
0
95
0
91
0
87
0
83
0
79
0
75
0
71
0
67
0
63
0
59
0
55
0
51
0
47
0
43
0
39
0
15
10
5
0
35
Frequency
Normal Distribution
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: exponential
distribution, mean = 20
Exponential Distribution
30
20
10
Random Numbers
76
71
66
61
56
51
46
41
36
31
26
21
16
11
6
0
1
Frequency
40
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