Engineering Data Analysis - Random Variables and Probability Distributions

These flashcards cover the fundamental definitions, formulas, and types of discrete and continuous probability distributions discussed in the Engineering Data Analysis lecture.

Random Variable

An item used to define or denote outcomes in the sample space; it assigns a numerical value to each outcome and its value cannot be known with certainty.

Discrete Random Variable

A type of random variable that can only assume a finite or countably infinite number of possible values, such as the number of customer arrivals or defective items.

Continuous Random Variable

A type of random variable that can take on any value within a given range, such as temperature, volume, weight, diameter, or time.

Probability Distribution Function (p.d.f.)

A table or function that help determine or compute the probability associated with each value of the random variable.

Expected Value ($E(x)$ or $\mu_x$)

The average of all possible values of the random variable $x$ or the mean of the $x$ values; it is a weighted average where probabilities represent the weights.

Expected Value Formula (Discrete)

$$E(x) = \sum x f(x)$$

Expected Value Formula (Continuous)

$$E(x) = \int x f(x) dx$$

States of Nature

In the decision-making process, these are the events that actually happen after a decision has been made.

Variance ($\sigma^2_x$)

A measure of the dispersion or spread of the values of $x$, calculated as the average of the squares of the deviations of all $x$ values from the mean.

Variance Working Formula

$$\sigma^2_x = E(x^2) - [E(x)]^2$$

Standard Deviation ($\sigma_x$)

The square root of the variance ($\pm (\sigma^2_x)^{1/2}$) which converts the variance into the same units as the random variable $x$.

Linearity Properties of Expectation

$E(ax + b) = aE(x) + b$ and $E[g(x) \pm h(x)] = E[g(x)] \pm E[h(x)]$ where $a$ and $b$ are constants.

Variance of a Constant ($\sigma^2_b$)

The variance of a constant $b$ is always equal to $0$.

Characteristics of Discrete Probability Distribution

1. $f(x) \ge 0$ for all $x$; 2. $\sum f(x) = 1$; 3. $P(X = x) = f(x)$.

Cumulative Distribution Function ($F(x)$, CDF)

A table or function that determines the probability that the random variable $X$ takes on values less than or equal to a specific value $x$, denoted as $F(x) = P(X \le x)$.

Discrete Uniform Distribution

A distribution where all values of the random variable have an equal chance of occurring, defined as $u(x:n) = \frac{1}{n}$.

Bernoulli Trial

A repeated trial in a binomial experiment where there are only two mutually exclusive outcomes: success ($p$) and failure ($q$).

Binomial Distribution

The probability distribution for the number of successes ($x$) in $n$ independent repeated trials, assuming sampling is done with replacement.

Multinomial Distribution

A general representation of the binomial distribution for experiments consisting of $n$ trials with more than two mutually exclusive outcomes per trial.

Negative Binomial Distribution

The probability distribution of the random variable $x$, representing the trial on which the $k^{th}$ success occurs.

Geometric Distribution

A case of the negative binomial distribution where the random variable $x$ represents the trial on which the first success occurs.

Hypergeometric Distribution

A distribution involving a sample of size $n$ selected without replacement from $N$ items, where items are classified as successes ($k$) or failures ($N-k$).

Poisson Distribution

A distribution used when an observer is concerned with the actual number of occurrences/successes per stated unit of time, space, volume, or area.

Poisson Approximation to Binomial

The Poisson distribution can approximate the binomial distribution when $n \ge 100$ and $p < 0.01$.

Binomial Approximation to Hypergeometric

A rule applied when the lot size $N$ is not given or is very large, allowing the binomial distribution to be used even if sampling is without replacement.