Discrete Probability Distributions

• The Probability Mass Function (PMF) is a function or table that maps every possible value $x$ of a discrete random variable to its probability $P(X = x)$.

• For a PMF to be valid, it must satisfy two fundamental conditions: every individual probability must be between 0 and 1 ($0 \le P(X=x) \le 1$), and the sum of all probabilities must equal 1.

The Summation Rule: $\sum_{all \, x} P(X = x) = 1$

• If a PMF is given as a formula containing an unknown constant $k$, the summation rule is used to solve for $k$ by setting the sum of all functional values to 1.

• The Cumulative Distribution Function (CDF), denoted as $F(x)$, represents the probability that the random variable $X$ takes a value less than or equal to $x$.

CDF Formula: $F(x) = P(X \le x) = \sum_{x_i \le x} P(X = x_i)$

• To calculate $P(X \le k)$, you sum the probabilities of all outcomes from the minimum possible value up to and including $k$.

• The Complement Rule is often used for efficiency: $P(X > k) = 1 - P(X \le k)$. This is particularly useful when calculating the probability of 'at least one' or large ranges.

• The Expected Value (or Mean), denoted as $E(X)$ or $\mu$, represents the long-term average outcome if the experiment were repeated many times.

Mean Formula: $E(X) = \sum x \cdot P(X = x)$

• Variance, denoted as $Var(X)$ or $\sigma^2$, measures the spread of the distribution around the mean. It is calculated as the expected value of the squared deviations from the mean.

Variance Formula: $Var(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \sum x^2 \cdot P(X = x)$

• The Standard Deviation ($\sigma$) is the square root of the variance and provides a measure of spread in the same units as the original random variable.

• In discrete distributions, the exact nature of the inequality matters significantly: $P(X < k)$ is not the same as $P(X \le k)$ because the value $k$ itself has a non-zero probability.

• A Discrete Uniform Distribution occurs when all $n$ possible outcomes are equally likely, resulting in $P(X=x) = \frac{1}{n}$ for every $x$.

• Sanity Check: Always verify that your probabilities sum to exactly 1. If they do not, check for arithmetic errors or missing outcomes in your sample space.

• Table Construction: For problems involving a small number of outcomes, always draw a probability table. It clarifies the relationship between $x$ and $P(X=x)$ and prevents missing values.

• Inequality Translation: Carefully translate phrases like 'at most' ($\le$), 'fewer than' ($<$), and 'at least' ($\ge$) into the correct mathematical symbols before calculating.

• Efficiency: If asked to find $P(X \ge 1)$, it is almost always faster to calculate $1 - P(X = 0)$ than to sum multiple individual probabilities.