Statistics and Probability - Introduction to Probability Distributions (Binomial and Normal)

Random Variables: A random variable is a numerical description of the outcome of a statistical experiment. Discrete random variables, like those in a Binomial distribution, take on countable values (e.g., $0, 1, 2$). Continuous random variables, like those in a Normal distribution, can take any value within an interval. Visually, discrete variables are often represented by distinct points or vertical bars on a graph, while continuous variables are represented by a smooth, unbroken line or curve.

The Binomial Distribution $X \sim B(n, p)$: This distribution models the number of successes in a fixed number of independent trials $n$, where each trial has only two possible outcomes (success or failure) and the probability of success $p$ remains constant. Visually, the distribution appears as a series of vertical bars (a histogram); if $p = 0.5$, the bars are perfectly symmetrical around the center, but if $p < 0.5$, the distribution is 'right-skewed' with a longer tail of bars extending to the right.

The Normal Distribution $X \sim N(\mu, \sigma^2)$: A continuous probability distribution characterized by its 'bell-shaped' curve. The curve is perfectly symmetrical around the mean $\mu$, which is the highest point of the graph. The total area under the curve is exactly $1$, representing the total probability. Visually, the curve approaches the horizontal axis at both ends but never touches it, showing that extreme values have very low but non-zero probabilities.

Standard Deviation and Curve Spread: In a Normal distribution, the standard deviation $\sigma$ dictates the 'fatness' or 'skinniness' of the bell curve. A small $\sigma$ results in a tall, narrow curve where data points are clustered closely around the mean $\mu$. A large $\sigma$ creates a short, wide, and flatter curve, indicating that the data is more spread out from the center.

The Empirical Rule (68-95-99.7 Rule): For any Normal distribution, approximately $68\%$ of the data falls within $1$ standard deviation of the mean ($\mu \pm \sigma$), $95\%$ falls within $2$ standard deviations ($\mu \pm 2\sigma$), and $99.7\%$ falls within $3$ standard deviations ($\mu \pm 3\sigma$). Visually, this is shown by shading sections of the bell curve: the middle section containing $68\%$ of the area is the most dense, while the 'tails' beyond $3\sigma$ contain very little area ($0.3\%$ combined).

Standard Normal Distribution and Z-scores: The Standard Normal Distribution is a special case where the mean $\mu = 0$ and standard deviation $\sigma = 1$, denoted as $Z \sim N(0, 1)$. A Z-score represents how many standard deviations a value $x$ is from the mean. On a graph, a positive Z-score is located to the right of the center, and a negative Z-score is to the left. This allows us to compare values from different normal distributions on a single standardized scale.

Cumulative Probability: For both distributions, the cumulative probability $P(X \le k)$ represents the probability that the variable takes a value less than or equal to $k$. In a Normal distribution, this is represented visually as the total shaded area under the curve to the left of the vertical line $x = k$. For a Binomial distribution, it is the sum of the heights of all bars from $x = 0$ up to $x = k$.