Statistics and Probability - Discrete Probability Distributions (Binomial, Poisson)

Discrete Random Variables (DRV): A DRV $X$ takes on a countable set of distinct values. The probability distribution is defined by a function $P(X=x)$ such that the sum of all probabilities equals $1$. Visually, this is represented by a probability mass function (PMF) where vertical bars at each $x$ value show the probability, and the total height of all bars is exactly $1$.

Expectation and Variance: The expected value $E(X)$ represents the theoretical mean or the 'balance point' of the probability distribution's histogram. The variance $Var(X)$ and standard deviation $\sigma$ measure the dispersion or 'spread' of the distribution around the mean; a larger variance results in a wider, flatter distribution visual.

Binomial Distribution Characteristics: A distribution is Binomial if it satisfies the BINS criteria: Binary outcomes (success/failure), Independent trials, fixed Number of trials $n$, and Same probability of success $p$. Visually, if $p=0.5$, the distribution is perfectly symmetrical; if $p<0.5$, it is right-skewed; and if $p>0.5$, it is left-skewed.

Binomial Parameters and Notation: Denoted as $X \sim B(n, p)$. The mean is located at $np$ and the variance is $np(1-p)$. When $n$ is large and $p$ is close to $0.5$, the discrete bar chart begins to approximate the bell-shaped curve of a normal distribution.

Poisson Distribution Characteristics: Models the number of events occurring in a fixed interval of time or space. It requires that events occur at a constant average rate $\lambda$, are independent, and do not occur simultaneously. Visually, for small $\lambda$, the distribution is highly right-skewed with the highest probability at $x=0$ or $x=1$, but as $\lambda$ increases, the peak moves to the right and the distribution becomes more symmetrical.

Poisson Parameters and Notation: Denoted as $X \sim Po(\lambda)$. A unique property of the Poisson distribution is that the mean and the variance are equal ($E(X) = Var(X) = \lambda$). This means the 'center' and the 'spread' of the distribution are linked to the same rate parameter.

Cumulative Probabilities: In discrete distributions, $P(X \le k)$ is the sum of probabilities for all outcomes from $0$ up to $k$. Graphically, this corresponds to the cumulative area of the bars to the left of and including the value $k$. It is important to distinguish between 'at most' ($P(X \le k)$), 'less than' ($P(X < k)$), and 'at least' ($P(X \ge k)$).

Poisson Approximation to the Binomial: When $n$ is very large ($n > 50$) and $p$ is very small ($p < 0.1$), the Binomial distribution $B(n, p)$ can be approximated by a Poisson distribution $Po(\lambda)$ where $\lambda = np$. This is visually useful as it simplifies calculations for rare events in large populations.