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Discrete Probability Distributions
Discrete Probability Distributions:
Bernoulli
Bernoulli Distribution
Definition
p: probability of success
X ~ Ber(p)
Conditions for a Bernoulli
There is one trial
There are two possible outcomes, success or failure
Formulae
P(X = x) = p
x
(1 - p)
1-x
x ∈ {0, 1}
P(X = 0) = 1 - p
P(X = 1) = p
E(X) = p
VAR(X) = p(1 - p)
Bernoulli Distribution
Probability of success: p
Bernoulli Distribution: X ~ Ber(0.5)
Parameter: p = 0.5
Statistic
Formula
Calculation
Value
E(X):
ΣxP(X = x)
1×0.5
0.5
Mean μ:
p
0.5
0.5
E(X
2
):
Σx²P(X = x)
1×0.5
0.5
Expected x
2
p
0.5
0.5
VAR(X):
E(X
2
) - E(X)
2
0.5 - 0.5
2
0.25
Variance σ
2
:
p(1 - p)
0.5 × 0.5
0.25
Standard Deviation σ:
√VAR(X)
√0.25
0.5
Statistic
Definition
Table of Values
Value
Median:
Minimum m such that
m
P(X = x)
Σ
x = 0
≤ 0.5
See Column:
P(X ≤ x)
0
Mode:
The value of x such that P(X = x) take its maximum value
See Column:
P(X = x)
0, 1
Bernoulli Distribution: X ~ Ber(0.5)
Mean: 0.5
Variance: 0.25
Bernoulli Distribution Calculation
Number of successful outcomes: x
Inverse Bernoulli Distribution Calculation
Cumulative probability: P(X ≤ x)
Start and end values for table of probabilities
Start Value:
End value:
Mode:
0, 1
Median:
0
x
P(X = x)
P(X < x)
P(X ≤ x)
P(X ≥ x)
P(X > x)
0
0.50000
0.00000
0.50000
1.00000
0.50000
1
0.50000
0.50000
1.00000
0.50000
0.00000
Binomial Distribution
Definition
n: number of trials
p: probability of success
X ~ B(n, p)
Conditions for a Binomial
The number of trials is fixed
The trials are independent of each other
There are two possible outcomes, success or failure
The probability of success is constant
Formulae
P(X = x) =
n
C
x
p
x
(1 - p)
n-x
=
n|
x! (n - x)!
× p
x
(1 - p)
n-x
E(X) = np
VAR(X) = np(1 - p)
Binomial Distribution
Number of trials: n
Probability of success: p
Geometric Distribution
Definition
p: probability of success
X ~ Geo(λ)
Conditions for a Geometric Distribution
The trials are independent of each other
There are two possible outcomes, success or failure
The probability of success is constant
Formulae
P(X = x) = p(1 - p)
x-1
P(X > x) = (1 - p)
x
P(X ≤ x) = 1 - (1 - p)
x
E(X) =
1
p
VAR(X) =
(1 - p)
p
2
Geometric Distribution
Probability of success: p
Negative Binomial Distribution
Definition
r: required number of successes
p: probability of success
X ~ NB(r, p)
Conditions for a Negative Binomial Distribution
The trials are independent of each other
There are two possible outcomes, success or failure
The probability of success is constant
Formulae
P(X = x) =
x-1
C
r-1
p
r
(1 - p)
x-r
=
(x - 1)|
(r - 1)! (x - r)!
× p
r
(1 - p)
x-r
E(X) =
r
p
VAR(X) =
r(1 - p)
p
2
NegativeBinomial Distribution
Number of successes: r
Probability of success: p
Poisson Distribution
Definition
λ: parameter
X ~ Po(λ)
Conditions for a Poisson Distribution
Each event is independent
Events occur at a constant average rate
Events cannot occur at the same time
Formulae
P(X = x) =
e
-λ
λ
x
x!
E(X) = λ
VAR(X) = λ
Poisson Distribution
Parameter: λ
Uniform Distribution
Definition
a: minimum value
b: maximum value
n: number of values = (b - a + 1)
X ~ U(a, b)
Conditions for a Discrete Uniform Distribution
All outcomes are evenly distributed (difference is one for this application)
All outcome are equally as likely
Formulae
P(X = x) =
1
n
E(X) =
a + b
2
VAR(X) =
n
2
- 1
12
Uniform Distribution
Minimum value: a
Maximum value: b
