Probability Distributions

1. Bernoulli Process

A Bernoulli process is a sequence of independent trials, each with two possible outcomes: success (1) or failure (0).

Probability of success: p
Probability of failure: q = 1 - p
Example: Tossing a coin (success = heads, failure = tails).

1. Bernoulli Distribution

The Bernoulli distribution describes a random experiment with two possible outcomes: success (with probability $p$ ) and failure (with probability $q = 1 - p$ ).

Mean = E (X) = p

Variance = Var (X) = p q

2. Binomial Distribution

The binomial distribution describes the number of successes in $n$ independent Bernoulli trials.

P (X = k) = C (n, k) \cdot p^{k} \cdot q^{n - k}

Where:

$C (n, k) = \frac{n!}{k! \cdot (n - k)!}$ is the combination formula.
$k$ = number of successes ( $0 \leq k \leq n$ ).

Properties:

Mean: $μ = n \cdot p$
Variance: $σ^{2} = n \cdot p \cdot q$

3. Negative Binomial Distribution

The negative binomial distribution describes the number of trials needed to achieve $r$ successes in a Bernoulli process.

P (X = k) = C (k - 1, r - 1) \cdot p^{r} \cdot q^{k - r}

Where:

$k$ = total number of trials ( $k \geq r$ ).
$r$ = number of successes.

Properties:

Mean: $μ = \frac{r}{p}$
Variance: $σ^{2} = \frac{r \cdot q}{p^{2}}$

4. Geometric Distribution

The geometric distribution describes the number of trials needed to achieve the first success in a Bernoulli process.

P (X = k) = q^{k - 1} \cdot p

Where:

$k$ = number of trials ( $k \geq 1$ ).

Properties:

Mean: $μ = \frac{1}{p}$
Variance: $σ^{2} = \frac{q}{p^{2}}$

5. Poisson Distribution

The Poisson distribution describes the number of events occurring in a fixed interval of time or space, given a constant average rate ( $λ$ ).

P (X = k) = \frac{e^{- λ} \cdot λ^{k}}{k!}

Where:

$k$ = number of events ( $k \geq 0$ ).
$λ$ = average rate of occurrence.

Properties:

Mean: $μ = λ$
Variance: $σ^{2} = λ$

6. Moment Generating Function (MGF)

The moment generating function (MGF) of a random variable $X$ is defined as:

M_{X} (t) = E (e^{t X})

Where:

$t$ = real number.
$E$ = expected value.

MGF of Distributions

Bernoulli Distribution:
$M_{X} (t) = q + p \cdot e^{t}$
Binomial Distribution:
$M_{X} (t) = {(q + p \cdot e^{t})}^{n}$
Negative Binomial Distribution:
$M_{X} (t) = {(\frac{p \cdot e^{t}}{1 - q \cdot e^{t}})}^{r}$
Geometric Distribution:
$M_{X} (t) = \frac{p \cdot e^{t}}{1 - q \cdot e^{t}}$
Poisson Distribution:
$M_{X} (t) = e^{λ (e^{t} - 1)}$

Comparison Table

Property	Binomial Distribution	Negative Binomial Distribution	Geometric Distribution
Random Variable	Number of successes in $n$ trials.	Number of trials to achieve $r$ successes.	Number of trials to achieve the first success.
Key Difference	Fixed number of trials ( $n$ ).	Fixed number of successes ( $r$ ).	Special case of negative binomial with $r = 1$ .
PMF	$P (X = k) = C (n, k) \cdot p^{k} \cdot q^{n - k}$	$P (X = k) = C (k - 1, r - 1) \cdot p^{r} \cdot q^{k - r}$	$P (X = k) = q^{k - 1} \cdot p$
MGF	$M_{X} (t) = {(q + p \cdot e^{t})}^{n}$	$M_{X} (t) = {(\frac{p \cdot e^{t}}{1 - q \cdot e^{t}})}^{r}$	$M_{X} (t) = \frac{p \cdot e^{t}}{1 - q \cdot e^{t}}$
Mean	$μ = n \cdot p$	$μ = \frac{r}{p}$	$μ = \frac{1}{p}$
Variance	$σ^{2} = n \cdot p \cdot q$	$σ^{2} = \frac{r \cdot q}{p^{2}}$	$σ^{2} = \frac{q}{p^{2}}$
Other Properties	Describes fixed trials.	Describes trials until $r$ successes.	Describes trials until the first success.

MCQ Questions

1. What is the mean of a binomial distribution with $n = 10$ and $p = 0.5$ ?

a) 5
b) 10
c) 2.5
d) 1

Answer: a) 5

2. Which distribution describes the number of trials until the first success?

a) Binomial
b) Geometric
c) Poisson
d) Negative Binomial

Answer: b) Geometric