Download Lecture 10 - Statistics

Today
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Today: Chapter 5
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Reading:
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Chapter 5 (not 5.12)
Suggested problems: 5.1, 5.2, 5.3, 5.15, 5.25, 5.33, 5.38, 5.47, 5.53, 5.62
Chapter 5
Continuous Random Variables
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Not all outcomes can be listed (e.g., {w1, w2, …,}) as in the case of
discrete random variable
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Some random variables are continuous and take on infinitely many
values in an interval
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E.g., height of an individual
Continuous Random Variables
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Axioms of probability must still hold
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Events are usually expressed in intervals for a continuous random
variable
0  P( E )  1; for any event E
P()  1
P( E )  P( F )  P( E )  P( F ) whenever E and F are mutually exclusive
Example
(Continuous Uniform Distribution)
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Suppose X can take on any value between –1 and 1
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Further suppose all intervals in [-1,1] of length a have the same
probability of occurring, then X has a uniform distribution on (-1,1)
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Picture:
Distribution Function of a Continuous Random Variable
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The distribution function of a continuous random variable X is
defined as,
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Also called the cumulative distribution function or cdf
Properties
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Probability of an interval:
Example
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Suppose X~U(-1,1), with cdf F(x)=1/2(x+1) for –1<x<1
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Find P(X<0)
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Find P(-.5<X<.5)
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Find P(X=0)
Example
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Suppose X has cdf,
 x / 3, if 0  x  1
F ( x)  
( x  1) / 3, if 1  x  2
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Find P(X<1/2)
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Find P(.5<X<3)
Distribution Functions and Densities
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Suppose that F(x) is the distribution function of a continuous random
variable
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If F(x) is differentiable, then its derivative is:
f ( x)  F ' ( x) 
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d
F ( x)
dx
f(x) is called the density function of X
Distribution Functions and Densities
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Therefore,
a
F (a) 
 f ( x)dx

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That is, the probability of an interval is the area under the density
curve
Example
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Suppose X~U(0,1), with cdf F(x)=x for –1<x<1
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What is the density of X?
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Find P(X<.33)
Properties of the Density
Example (5-16)
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Suppose X is a random variable and it is claimed that X has density
f(x)=30x2(1-x)2 for 0<x<1
Is f(x) a density?
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If yes, find the c.d.f. of X.
Example (5-15)
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Suppose X is a random variable and X has density f(x)=c(1-|x|) for
|x|<1 and c is a positive constant
Find c?
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Draw a picture of f(x)
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Find P(X>1/2)
Example
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X has an exponential density:

e x if x  0
f ( x)  

 0 otherwise
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Find F(x)
Example
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X has an exponential density:

e x if x  0
f ( x)  

 0 otherwise
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Find the density of Y=X1/2
Transformations
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If Y=g(x) is a one-to-one function with inverse, g-1(x), the density of
Y can be obtained from the density of X as,
Example
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X has an exponential density:

e x if x  0
f ( x)  

 0 otherwise
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Find the density of Y=X1/2
Example (5-21)
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Suppose X~U(-1,1)
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Find the density of Y=|X|
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Find P(-.5<Y<.75)