Unformatted text preview: 2 Continuous Random Variables 3 Probability Distributions and
Probability Density Functions 4 Probability Density Function
y
y
Defined 5 Probability Density Functions
and Histograms 6 A Probability Density Function
Example
Example 41
E
l 41
Let the continuous random variable X denote the current measured in a thin copper wire
p
g
in milliamperes. Assume that the range of X is [0, 20 mA], and assume that the
probability density function of X is f(x) = 0.05 for 0 x 20. What is the probability
that a current measurement is less than 10 milliamperes?
The
Th probability density function is shown in the figure below (it is assumed that f( ) =
b bili d i f
i i h
i h fi
b l (i i
d h f(x)
0 wherever it is not specifically defined). The probability requested is indicated by the
shaded area in the figure, which can be computed using the equation from slide #5.. 7 Another Probability Density
y
y
Function Example
Example 42
p 8 The Cumulative Distribution
Function in the Continuous Case 9 A Cumulative Distribution
Function Example Example 44
p 10 Cumulative Distribution Function
Examples
• Here is the cdf for
Example 42: • Here is the cdf for
Example 41: 11 Mean and Variance of a
Continuous Random Variable 12 Example of Mean and Variance of a
Continuous Random Variable
Example 46 13 The Mean of a Function of a
Continuous Random Variable 14 The Continuous Uniform
Distribution 15 The Mean and Variance of the
Continuous Uniform Distribution • Use these form la for the situation in
formula
sit ation
example 41 (and 46) to verify the results
from slide #13 that when a=0 and b=20:
ȝ = 10.0 and ı2 = 33.33
16 The cdf of the General
Continuous Uniform Distribution 17 The Normal Distribution 18 Example Normal Distribution
pdfs 19 WellKnown Normal
Distribution Probabilities
• F any normal random variable:
For
l
d
i bl 20 The Standard Normal
Distribution 21 How to Use a Table of the cdf of
the Standard Normal Distribution
Example 411
4 11 Figure 413 Standard normal probability density function. 22 Using Th St d d N
U i The Standard Normal cdf to Determine
l df t D t
i
Probabilities for Other Normal Distributions 23 An Example on Standardizing a
p
g
Normal Random Variable
Example 413
4 13 24 Another Example (414) on Standardizing A
p
g
Normal Random Variable 25 The Normal Approximation to
the Binomial Distribution 26 An Example (
p (417 & 418) of a Normal
)
Approximation to the Binomial 27 Guidelines for Using the Normal
to Approximate the Binomial or
pp
Hypergeometric 28 The Normal Approximation to
the Poisson Distribution 29 An Example of a Normal
Approximation to the Poisson
Example 420 30 The Exponential Distribution The exponential distribution cdf is:
O
F(x) = 1eOx 31 An Exponential Distribution
Example (Example 421) 32 An Exponential Distribution
Example (Example 4 21 continued)
421 33 An Exponential Distribution
p
Example (Example 421 continued) 34 An Unusual Property of the
Exponential Distribution
Lack of Memory Property E.g.
In Example 421, suppose that there are no logons from
12:00 to 12:15; the probability that there are no logons from
12:15 to 12:21 is still 0 082 Because we have already been
0.082.
waiting for 15 minutes, we feel that we are “due.” That is, the
probability of a logon in the next 6 minutes should be
greater than 0.082. H
t th 0 082 However, f an exponential
for
ti l
distribution this is not true. 35 The Erlang Distribution
The random variable X that equals the interval length until r
h
d
i bl
h
l h i
ll
h
il
counts occur in a Poisson process with mean Ȝ > 0 has and
Erlang random variable with parameters Ȝ and r The
r.
probability density function of X is for
f x > 0 and r =1, 2, 3, ….
d
36 Example pdfs for the Erlang
Distribution
Misc. Erlang pdfs for different values of r (the "order") {O=0.5}
1.0
10
0.9
0.8
0.7
0.6
0.5
05
0.4
0.3
0.2
0.1
0.0
0.0 1.0 2.0
r=1 3.0
r=2 4.0
r=4 5.0 6.0 7.0 8.0 r=8 37 The Gamma Distribution 38 Example pdfs for the Gamma
Distribution
Figure 425 Gamma
probability density
functions for selected
values of r and O. 39 The Weibull Distribution 40 Example pdfs for the Weibull
Distribution Figure 426 W ib ll
Fi
4 26 Weibull
probability density
functions for selected
values of D and E. 41 The Lognormal Distribution 42 Example pdfs for the Lognormal
Distribution 43 44 ...
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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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