Outline Discrete Probability Distributions The Binomial - - PDF document
12/7/2006 219323 Probability y and Statistics for Software and Knowledge Engineers Lecture 4: Discrete and Continuous Probability Distributions Monchai Sopitkamon, Ph.D. Outline Discrete Probability Distributions The Binomial
Monchai Sopitkamon, Ph.D.
Discrete Probability Distributions
Continuous Probability Distributions
Bernoulli RVs
If n independent Bernoulli trials are
X ∼ B(n, p) counts the number of
Pmf of a B(n, p) RV:
x n x
−
E(X) = E(X1)+…+ E(Xn) = p +…+ p = np Var(X) = Var (X1) +…+ Var(Xn) = p(1-p)
Symmetric Prob Distribution B(n, 0.5) with respected to E(X)=n/2
P(X≥6) = 1-P(X≤5) = 1-0.855 = 0.145
Figure 3.2 Probability mass function and cumulative distribution
function of a B(8, 0.5) random variable
5 3 5 3
3 2 3 1 ! 5 ! 3 ! 8 3 1 1 3 1 3 8 ) 3 ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = X P
= 0.273 P(X≤1) = P(X=0) + P(X=1) = 0.039 + 0.156 = 0.195 Figure 3.3 Probability mass function and cumulative distribution
function of a B(8, 1/3) random variable
P(X=4) = 0.171
P(X≤2) = P(X=0) + P(X=1) + P(X=2) = 0 001 + 0 002 + 0 017 0.001 + 0.002 + 0.017 = 0.020 Figure 3.4 Probability mass function and cumulative distribution
function of a B(8, 2/3) random variable
8
8 0.82 0.18 8 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
8
8 0.82 0.18 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠
0.2044 ≈ 0.0000011 ≈
6 2 7 1 8
8 8 8 0.82 0.18 0.82 0.18 0.82 0.18 6 7 8 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
0.2758 0.3590 0.2044 ≈ + +
Discrete Probability Distributions
Continuous Probability Distributions
The number of trials up to and including
The pmf P(X = x) = (1 – p)x-1p for x = 1,
x 1
−
The cdf
x
2
Prob getting head the first time after 3 failures = P(X=4) = (1-p)4-1p
3
= (1/2)3x(1/2) = 1/16
Figure 3.11 Probability mass function and cumulative distribution
function of a geometric distribution with parameter p = 1/2
r r x
−
2
Modeling number of tosses of a fair coin until until two heads are obtained.
1 5 5 1 ) 2 (
2
⎟ ⎞ ⎜ ⎛ X P 4 5 . 5 . 1 ) 2 (
2
= × × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = X P
Figure 3.13 Probability mass function and cumulative distribution function of a negative binomial distribution with parameters p = ½ and r = 2
138 . 6 . 4 . 2 5 ) 6 (
3 3
= × × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = X P
Figure 3.16 Probability mass function of a negative binomial distribution with
parameters p = 0.6 and r = 3, the distribution of the number of applicants interviewed
Discrete Probability Distributions
Continuous Probability Distributions
Used when selecting a group of identical r
If n items are chosen at random without If n items are chosen at random without
We have a hypergeometric distribution of the
The pmf is
for max(0, n+r-N) ≤ x ≤ min(n, r) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = n N x n r N x r x X P ) (
⎟ ⎞ ⎜ ⎛ − × × × ⎟ ⎞ ⎜ ⎛ − = r r n n N X Var 1 ) ( ⎟ ⎠ ⎜ ⎝ × × × ⎟ ⎠ ⎜ ⎝ − N N n N X Var 1 1 ) (
412 . ! 11 ! 5 ! 16 ! 7 ! 3 ! 10 ! 4 ! 2 ! 6 5 16 3 10 2 6 ) 2 ( = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ × ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = X P
Figure 3 .1 7 Probability m ass function of the num ber of underw eight m ilk containers in the inspector’s sam ple
875 . 1 16 6 5 ) ( = × = = N nr X E
Discrete Probability Distributions
Continuous Probability Distributions
Used to model the number of times that
x
λ −
Figure 3.19 Probability mass function and cumulative distribution function of a Poisson random variable with mean λ = 2.0 Figure 3.20 Probability mass function and cumulative distribution function of a Poisson random variable with mean λ = 5.0
180 . 6 8 135 . ! 3 2 ) 3 (
3 2
= × = × = =
−
e X P
125 . 2 25 1 5 1 1 ! 2 5 ! 1 5 ! 5 ) 2 ( ) 1 ( ) ( ) 2 (
5 2 5 1 5 5
= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = × + × + × = = + = + = = ≤
− − − −
e e e e X P X P X P X P
3 3
− −
577 . 423 . 1 2 9 1 3 1 1 1 ! 2 3 ! 1 3 ! 3 1 ) 2 ( ) 1 ( ) ( 1 ) 3 (
3 2 3 1 3 3
= − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + − = × − × − × − = = − = − = − = ≥
− − − −
e e e e X P X P X P X P
Figure 3.21 Probability mass function and cumulative distribution function
number of software errors