Download Math 4030-2c Discrete Random Variables, Probability, and

Math 4030-3a
(Discrete) RV
Binomial
Hypergeometric
Random variable (Sec. 4.1)
A function that assigns a numerical value to
each possible outcome in the sample space.

Sample space S


One value for one
outcome. i.e. different value
must mean different
outcomes.
However, different
outcomes may have the
same value.
Random variable may take
discrete or continuous
values.
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Probability Distribution of a
discrete random variable:
A list of probability values corresponding
to all values of a discrete random variable
X. i.e.
f ( x)  P[ X  x],
for any value x that the random variable
X takes.
f ( x)  0, for all x.
 f x  1.
all x
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• Probability histogram and bar chart;
• Cumulative distribution function F(x):
F ( x)  P[ X  x]   f ( y ).
y x
If X takes values
x1  x2    xn ,
then
f ( xi )  P[ X  xi ]  F xi   F xi 1 .
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Binomial distribution (Sec. 4.2)
The experiment consists n trials, each
have two outcomes: S or F (Bernoulli
Trials)
2. Probability of success are the same in
each Bernoulli trial, say p.
3. The n trials are independent.
Let X = number of successes in n trials.
1.
X ~ Bi (n, p)
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The probability distribution of X:
n x
P( X  x )  b( x; n, p )    p (1  p )n  x ,
 x
x  0,1,2,..., n.
Cumulative distribution:
x
P ( X  x )  B ( x; n, p )   b( k ; n. p ),
k 0
x  0,1,2,..., n.
Table 1 on Page 513.
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Properties:
• Binomial numbers
• Symmetry of probabilities
Applications:
Repetition of independent binary
experiments (coin toss), and count the
number of one outcome (number of
head)
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Mean and Variance:
  EX   np,
  V X   npq  np1  p .
2
Sampling with or without replacement:
With replacement:
Without replacement:
X ~ H (n, a, N )
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Hypergeometric Distribution (Sec.
4.3)
There are N units, of which a units are
defective.
Randomly sample n units without
replacement, and let X be the number of
defective units in the sample.
Then
X ~ H (n, a, N )
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Probability P(X=x) = h(x; n, a, N)?
 a  N  a 
 

x  n  x 

h( x; n, a, N ) 
.
N
 
n
Mean:
Variance:
a
  EX   n 
N
a 
a   N n
  n   1    

N  N   N 1 
2
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N-a
Binomial vs.
Hypergeometric
n-x
x
a





In an infinitely large pool,
p100% are marked with
an “S”.
Randomly select n.
X is the number of S’s in
the sample of n.
X ~ Bi(n, p)
All possible values for X:
X = 0, 1, 2, …, n



In a pool of N objects, a are
marked with an “S”.
Randomly select n.
X is the number of S’s in the
sample of n.
X ~ H(n, a, N)
All possible values for X:
0xn
0xa
0n–xN-a
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Binomial vs. Hypergeometric
X ~ Bi(n, p)
X ~ H(n, a, N)
P( X  x)  b( x; n, p)
P( X  x)  h( x; n, a, N )
n x
n x
   p 1  p 
 x
 a  N  a 
 

x  n  x 


N
 
n
x  0,1,..., n
0  x  n, 0  x  a
0 n x  N a
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