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Unformatted text preview: 2 Discrete Random Variables 3 A Simple Discrete Random
Variable Example
Example 31
p 4 Probability Distributions and
Probability Mass Functions Figure 31 Probability distribution for bits in error.
5 Probability Mass Function
Defined 6 A Probability Mass Function
Example
Example 35 7 A pmf Example (continued) 8 Cumulative Distribution Function
Defined 9 A Cumulative Distribution
Function Example
• Determine the probability mass function of X from the
following cumulative distribution function: • From either the graph or the function definition, we can see
that the pmf is:
I I I
10 Mean,Variance, and Standard
Deviation of a Discrete Random
Variable 11 Example of Mean and Variance
of a Discrete Random Variable
Example 311 12 The Mean of a Function of a
Discrete Random Variable
• We are often interested in some function of a
random variable X
– Denoting the function of interest as h(X): • E.g. as a selfstudy exercise, confirm that for
( )
a d o
o t e p ev ous s de:
h(X)=X2 and for X as on the previous slide:
– E[h(X)] = 158.1
13 The Discrete Uniform
Distribution
• • E.g., here’s the pmf of a discrete uniform
distribution i h
di ib i with range 0 to 9 (i l i )
(incluive) 14 Mean and Variance of The
Discrete Uniform Distribution
• • E.g., for the example on the previous slide
D and E yielding:
– ȝ = 4.5 and ı2 = 8.25 15 The Binomial Distribution 16 Examples of Random Experiments That Might Fit
l
f
d
i
h
i h i
The Binomial Distribution Assumptions 17 Example pmfs for the Binomial
Distribution 18 A Numeric Binomial Distribution
Example
Example 318 19 A Numeric Binomial Distribution
Example Example 318 (continued) 20 The Mean and Variance of the
Binomial Distribution • E g for Example 318 earlier we know that
E.g.,
E ample 3 18
e kno
Q and S yielding:
– ȝ = 1.8 and ı2 = 1.62
21 The Geometric Distribution 22 A Geometric Distribution
Example Example 320 • Also, we can compute the mean and variance using the
formulae on the previous slide (with S ) yielding:
) – ȝ = 10 and ı2 = 90 23 Example pmfs for the Geometric
Distribution 24 An Unusual Property of the
Geometric Distribution
Lack of Memory Property 25 The Negative Binomial
Distribution 26 Example pmfs for the Negative
Binomial Distribution 27 The R l ti hi B t
Th Relationship Between The Geometric and
Th G
ti
d
Negative Binomial Distributions Figure 311. Negative binomial random variable
represented as a sum of geometric random variables
variables.
28 A Negative Binomial
Distribution Example
Example 325 29 A Negative Binomial
Distribution Example
Example 325 (continued) 30 The Hypergeometric Distribution 31 A Hypergeometric Distribution
Example
• Modifying Example 318, suppose we have a batch of 50 samples of
water, 5 of which are contaminated
• If we draw a random sample of size 2, without replacement, from these
50, what’s the distribution of the number of contaminated samples?
N 50, K 5,
n 2
• This is a hypergeometric random variable with N=50 K=5 and n=2
(not a binomial random variable with p=0.1 and n=2)
• Using equations 313 and 37 to determine pmf values yields: f(x)
x
0
1
2 Hypergeometric
yp g 80.82%
18.37%
18 37%
0.82% Binomial
81.00%
18.00%
18 00%
1.00% 32 Example pmfs for the
Hypergeometric Distribution
Figure 312.
Hypergeometric
distributions for
selected values of
parameters N, K, and n. 33 Another Hypergeometric
Distribution Example
Example 327 34 Another Hypergeometric
Distribution Example
Example 327 (continued) 35 Mean and Variance of The
Hypergeometric Distribution • For Example 327 these formulae yield:
– ȝ = 1.33
and
ı2 = 0.88
36 Binomial and Hypergeometric
Distributions Compared
• The mean for each distribution is the same (if p is
interpreted as the proportion of “successes” in the
i
d
h
i
f“
”i h
whole batch)
• The variance differs only in a multiplication
factor in the case of the hypergeometric: 37 Binomial and Hypergeometric
pmfs Compared Figure 313. Comparison of hypergeometric and binomial
distributions. 38 The Poisson Distribution 39 A Poisson Distribution Example
Example 333 40 Mean and Variance of The
Poisson Distribution • Example pmfs for this distribution:
Probability Mass Functions of the
Poisson Distribution 30.0% 25.0% 20.0% O
O 15.0% 10.0% 5.0% 0.0%
0 2 4 6 8 10 12 14 16 18 20 41 42 ...
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This note was uploaded on 05/31/2010 for the course ENGG 319 taught by Professor Nanayakkara during the Fall '08 term at University of Calgary.
 Fall '08
 NANAYAKKARA

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