illustrative abstractions

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# Discrete Random Variable

Given $X$, a discrete random variable,

Population Mean (Expectancy)

Population Variance

If two discrete random variables $X$ and $Y$ are independent,

# Bernoulli Distribution

If success, $X = 1$. If failure, $X = 0$.

# Binomial Distribution

$X =$ number of successes in $n$ independent repetitions of the experiment.

Define $X_i$ as the $i$th repetition’s success or not. $X_i$ follows a Bernoulli Distribution.

# Geometric Distribution

$X =$ number of tries up to and including first success

Let

Given $0 < q < 1, \quad n \xrightarrow{} \infty$

Such that $E(X) = \frac{p}{(1 - q)^2} = \frac{1}{p}$

Using the same method as above, $E(X^2) = p\frac{1 - q^2}{(1 - q)^4} = p\frac{1 + q}{(1 - q)^3} = \frac{1 + q}{(1 - q)^2}$

# Poisson Distribution (To be continued…)

Poisson Distribution is essentially a binomial distribution with very large $n$ (very small time interval).