Download 1 Recap: Discrete Distributions What is ahead: Continuous

Recap: Discrete Distributions
r N r
 

Hypergeometric P( X  x)   x   n  x  , x  max{0, n  ( N  r)}, ,min{n, r}
N
 
n
E( X )  np, where p  r / N .
 
Var( X )  fpc  n  p 1  p 
 x 1
xr r
P( X  x)  
 (1  p) p , x = r, r+1, …
Negative Binomial
 r 1
E( X ) 
r
p
V (X ) 
section 3.6
r(1 p)
p2
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What is ahead: Continuous Distributions
• Generic setup for continuous distributions:
• Cumulative Distribution Function (cdf): F(x) = P(X ≤ x)
• Probability Density Function (pdf) : f(x) “similar” to pmf
• P (X ≤ b), P(X ≥ a), P(a ≤ X ≤ b), ….
• Expectation : E[X] = µ
• Variance : V(X) = E [ (X - µ)2 ]
• Specific Continuous Distribution
• Uniform
• Normal (Gaussian)
• Gamma, Exponential, Chi-squared, …
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Continuous Random Variables
Sections 4.1, 4.2
Recall: X is a continuous random variable if the set of its possible values
is an interval (finite or infinite) or union of finite intervals).
Reality...
A continuous random variable is used as a model for a quantity that,
in principle, takes values in an interval (subset of the real line), even
though every measurement scale is, in truth, discrete.
Example: The pH of a chemical compound is measured (with error).
Possible values are X  [0,14], however the measuring device only
reads to a two decimal places. We would still treat X as a continuous
random variable.
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Continuous Random Variables
Idea: Use calculus to replace sums (∑) with integrals ().
Warning: There are important differences between discrete/continuous RVs!
Specifying the Probability Distribution of a RV
Discrete RV
Continuous RV
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Cumulative Distribution Function and
Probability Density Function
Def: The cumulative distribution function (cdf) FX(x) for a continuous
random variable X is defined for every real x by
FX ( x)  P( X  x),   x  .
(i)
F ( x) is a non-decreasing function, with
(ii)
d
F ( x) exists
dx
lim F ( x)  1 and lim F ( x)  0.
x
x
everywhere except at a finite number of points.
(iii) Since F is non-increasing,
d
F ( x)  f X ( x)  0
dx
(iv) Fundamental Theorem of Calculus 
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Probability Density Functions
Def: A function fX : R  [0, ∞) is a probability density function (pdf) for
a continuous random variable X if for any a ≤ b,
Notes
(a) Graphical Interpretation:
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CDF/PDF Relationship
(a) cdf  pdf
Given a cdf F(.)
find the pdf f(.)
(b) pdf  cdf
Given a pdf f(.)
find the cdf F(.)
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Probability Density Functions
(b) Notation:
(c) Every pdf f(·) satisfies:
(d) Every continuous RV X satisfies:
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Dart Board Example
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CDFs / PDFs
(g) P (X < b) =
10
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CDFs / PDFs
(h) P (X ≥ a) =
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CDFs / PDFs
(i) P (a < X ≤ b) =
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CDFs / PDFs
(j) P (X ≤ a OR X > b) =
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pmf’s vs. pdf’s
pmf p(x)
pdf f(x)
Interpretation
Bounds
Total
Ex:
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Example pdf and cdf for continuous RVs
4.12 The cdf for a random variable X representing error in a particular
measurement is
0

 1 3
FX ( x)    (4x  x3 3)
 2 32
1

x  2
2  x  2
x2
Find the pdf of X.
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Example pdf and cdf for continuous RVs
Show how to use both the pdf and cdf to compute P(X > 0).
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Example pdf and cdf for continuous RVs
Find P(X < 0.5 or X > 0.5).
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The Uniform Distribution
Def: A continuous RV X is said to have a uniform distribution on
the interval [A , B] if the pdf of X is
Notation
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The Uniform Distribution
Example: Say that the # of minutes past 10:27 that a student arrives
at this class is uniformly distributed over [0 , 8]. (Reasonable?)
Percentiles of a Continuous Random Variable
Say a person’s test score was at the 85th percentile...what does this mean?
Let p be between 0 and 1. The (100 × p)th percentile of the
distribution of a continuous random variable X, denoted by (p), is
defined by
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pdf / cdf
Median of a Continuous RV
The median of a continuous random variable is the 50th percentile of
the distribution:
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Expected Values of a Continuous RV
Recall: X discrete with pmf pX(x):
( = long run average
as n  ∞)
Definition: If X is a continuous RV with pdf fX(·):
Special Cases
(a) Mean of X  E[ X ]
E [X2] =
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Special Cases (contd.)
(b) Variance:
Properties and Warnings!
All the properties of expected values that we discussed for discrete
random variables apply to continuous random variables!
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4.12 (contd.) The cdf for a random variable X representing error in a
particular measurement is
0

 1 3
FX ( x)    (4x  x3 3)
 2 32
1

x  2
2  x  2
x2
What is the median?
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What is the probability that X lies between two standard deviation of its
mean?
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