DADM Chapter 5: Probability Distributions

Vocabulary flashcards covering weighted sums of normal random variables, the binomial distribution and its sampling rules, and the Poisson distribution.

Weighted Sum of Normal Random Variables

A linear combination of normally distributed random variables ($a_1 X_1 + \dots + a_n X_n$) that is also normally distributed, holding true even if the individual random variables are not independent.

Sum of Independent and Identically Distributed Normal Random Variables

The sum $X_1 + \dots + X_n$ of $n$ independent normal random variables sharing common mean $\mu$ and common standard deviation $\sigma$, which is normally distributed with mean $n\mu$, variance $n\sigma^2$, and standard deviation $\sqrt{n}\sigma$.

Mean and Variance of a General Weighted Sum of Normal Variables

For constants $a_1$ through $a_n$, the weighted sum $a_1 X_1 + \dots + a_n X_n$ has a mean of $a_1\mu_1 + \dots + a_n\mu_n$ and a variance of $a_1^2\sigma_1^2 + \dots + a_n^2\sigma_n^2$.

Binomial Distribution

A discrete probability distribution representing the random number of successes $X$ across $n$ independent, identical trials, characterized by parameters $n$ (number of trials) and $p$ (probability of success on each trial).

Conditions for a Binomial Distribution

Situations that occur when sampling from a population with only two types of members, or when performing a sequence of identical experiments where each trial has only two possible outcomes with constant success probability $p$ and failure probability $1 - p$.

Mean of the Binomial Distribution

The expected value of a binomial random variable with parameters $n$ and $p$, defined by the formula $E(X) = np$.

Standard Deviation of the Binomial Distribution

The measure of dispersion for a binomial random variable with parameters $n$ and $p$, defined by the formula $\text{Stdev}(X) = \sqrt{np(1 - p)}$.

Empirical Rules for the Binomial Distribution

Approximations stating that there is about a $95\%$ chance that the actual number of successes falls within two standard deviations of the mean, and almost no chance that it falls more than three standard deviations from the mean.

Sampling Without Replacement

A sampling method in which each member of the population can be sampled only once; once an individual is selected, their name is struck from the list and cannot be selected again.

Sampling With Replacement

A sampling method where selected members remain available in the population, making it possible to select a given member more than once across trials.

10% Rule for Binomial Sampling

A guideline stating that if no more than $10\%$ of a population is sampled, the binomial model can safely be used even when sampling is conducted without replacement.

Poisson Distribution

A discrete probability distribution that models the number of events occurring within a specified period of time or space, taking nonnegative integer values ($0, 1, 2, \dots$) without an upper limit.

Parameter Lambda ($\lambda$)

The single, strictly positive parameter of a Poisson distribution, commonly interpreted as an arrival rate (such as arrivals per hour), which simultaneously represents both the mean and the variance of the distribution.