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) = px(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(X2):

Σx²P(X = x)

1×0.5

0.5

Expected x2

p

0.5

0.5

VAR(X):

E(X2) - E(X)2

0.5 - 0.52

0.25

Variance σ2:

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) = nCx px(1 - p)n-x
=
n|
x! (n - x)!
× px(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)
p2
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-1Cr-1pr(1 - p)x-r
=
(x - 1)|
(r - 1)! (x - r)!
× pr(1 - p)x-r
E(X) =
r
p
VAR(X) =
r(1 - p)
p2
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-λλx
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) =
n2 - 1
12
Uniform Distribution

Minimum value: a

  

Maximum value: b