Bernoulli Distribution

The fundamental building block of discrete probability, modeling a single trial with two outcomes.

The "Single Coin Flip"

The Bernoulli distribution is the simplest of all discrete distributions. It models a single event or trial that has only two possible outcomes: a "success" or a "failure".

Think of it as a single coin flip (Heads or Tails), a single trade (Win or Loss), or a single bond (Default or No Default). The entire distribution is described by a single parameter, $p$ , which is the probability of success.

Interactive Bernoulli Trial

Adjust the probability of success (

p

) to see how it affects the outcome probabilities.

Probability of Success (p): 0.70

Mean (

\mu

): 0.70

Variance (

\sigma^2

): 0.21

Core Concepts

Probability Mass Function (PMF)

The PMF gives the probability that the random variable

X

is exactly equal to some value

k

. For a Bernoulli trial,

k

can only be 0 (failure) or 1 (success).

P(X=k) = p^k (1-p)^{1-k} \quad \text{for } k \in \{0, 1\}

If $k=1$ (success), the formula becomes $P(X=1) = p^1(1-p)^{1-1} = p \cdot 1 = p$ .
If $k=0$ (failure), the formula becomes $P(X=0) = p^0(1-p)^{1-0} = 1 \cdot (1-p) = 1-p$ .

The interactive chart above is a direct visualization of this PMF.

Cumulative Distribution Function (CDF)

The CDF gives the probability that the random variable

X

is less than or equal to some value

x

. It's a running total of the PMF.

F(x) = P(X \le x)

Value of $x$	CDF: $F(x) = P(X \le x)$	Explanation
$x < 0$	$0$	The outcome cannot be less than 0.
$0 \le x < 1$	$1-p$	The only possible value in this range is 0.
$x \ge 1$	$1$	Includes both outcomes 0 and 1, so probability is $(1-p) + p = 1$ .

Key Derivations

Deriving the Mean & Variance

Deriving the Expected Value (Mean)

The expected value is the sum of each outcome multiplied by its probability.

Step 1: Set up the Summation

The formula for the expected value of a discrete random variable is:

E[X] = \sum_{k} k \cdot P(X=k)

Step 2: Apply to the Bernoulli Case

For the Bernoulli distribution, our outcomes ( $k$ ) are 0 (failure) and 1 (success).

E[X] = (0 \cdot P(X=0)) + (1 \cdot P(X=1))

E[X] = (0 \cdot (1-p)) + (1 \cdot p)

E[X] = 0 + p

Step 3: Final Result

Final Mean Formula

E[X] = p

This makes intuitive sense: if a trade has a 70% ( $p=0.7$ ) chance of success, the expected outcome of a single trial is 0.7.

Deriving the Variance

We use the formula $Var(X) = E[X^2] - (E[X])^2$ . We already know $E[X] = p$ , so we first need to find $E[X^2]$ .

Step 1: Find the Second Moment, E[X²]

We use the same summation logic, but with $k^2$ .

E[X^2] = \sum_{k} k^2 \cdot P(X=k)

E[X^2] = (0^2 \cdot P(X=0)) + (1^2 \cdot P(X=1))

E[X^2] = (0 \cdot (1-p)) + (1 \cdot p) = p

Step 2: Calculate the Variance

Now, we substitute everything back into the variance formula:

Var(X) = E[X^2] - (E[X])^2 = p - p^2

Final Variance Formula

Var(X) = p(1-p)

Notice that the variance is maximized when $p=0.5$ (a 50/50 coin flip has the highest uncertainty) and is 0 when $p=0$ or $p=1$ (the outcome is certain).