SCM EXAM 2 Chapter 5: Discrete Probability Distributions

Vocabulary practice flashcards covering random variables, expected value, variance, standard deviation, discrete uniform distributions, Bernoulli processes, the Binomial distribution, the Poisson distribution, and their associated Excel formulas from Chapter 5.

What is a Random Variable and some examples

A function that turns every possible outcome of an experiment into a number, making it easy to measure uncertainty and calculate probabilities.

• captures uncertainty

• summarizes outcomes of an experiment with numerical values

EX:

Number of customer orders in a day

what are the 2 types of a random variable

Discrete and continuous.

Discrete Random Variable

A random variable that assumes a countable number of distinct values denoted as $x_1, x_2, \dots, x_n$, such as the number of employees in a firm.

Continuous Random Variable

A random variable characterized by uncountable values in an interval that cannot be summarized with a discrete list, such as the return on a mutual fund.

every discrete random variable is associated with what

A PROBABILITY DISTRIBUTION

Discrete Probability Distribution

A distribution that provides the probability $P(X = x)$ that a random variable $X$ assumes a particular value $x$, satisfying the properties $0 \le P(X = x) \le 1$ and $\sum P(X = x_i) = 1$.

Different ways to represent the probability distribution

EX: The probability of rolling a die

• discrete uniform distribution

• finite number of values

• each value is equally likely to occur

• symmetric

what is the Expected Value of a distribution ($E(X)$ )

• also referred to as the mean, weighted average of all possible values of X

• measure of central location

• denoted as $E(X)$ or $\mu$

• calculated as $E(X) = \mu = \sum x_i P(X = x_i)$.

what is Variance ($Var(X)$)

A measure of dispersion(the spread/scatter of values in a probability distribution) indicating whether the values of $X$ are clustered about $\mu$ or widely scattered from $\mu$, denoted as $Var(X)$ or $\sigma^2$, and calculated as $Var(X) = \sigma^2 = \sum (x_i - \mu)^2 P(X = x_i)$.

different types of dispersion

• Low dispersion: The possible values of X are tightly clustered close to the average or expected value (μ).

• High dispersion: The possible values of X are widely scattered and spread far apart from the average (μ).

what is the Standard Deviation ($SD(X)$) how is it denoted and calculated

The square root of the variance used to measure dispersion around the mean, denoted as $SD(X)$ or $\sigma$, and calculated as $SD(X) = \sigma = \sqrt{\sigma^2}$.

Bernoulli Process

A type of experiment consisting of a series of $n$ independent and identical trials where each trial has ONLY TWO possible outcomes (success and failure) and the probability of success ($p$) and failure ($1 - p$) remain constant across trials.

what is a Binomial Random Variable

A discrete random variable defined as the number of successes achieved in $n$ trials of a Bernoulli process, taking possible integer values $0, 1, \dots, n$.

Binomial Probability Distribution Formula

The formula used to calculate the probability of $x$ successes in $n$ Bernoulli trials: $P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} = \frac{n!}{x!(n - x)!} p^x (1 - p)^{n - x}$ for $x = 0, 1, 2, \dots, n$, where $0! = 1$.

what are the two parts of the Binomial Probability Distribution Formula

• FIRST TERM $\binom{n}{x} = \frac{n!}{x!(n - x)!}$ tells how many sequences with X successes and n-x failures are possible in n trials

• SECOND TERM $p^x (1 - p)^{n - x}$ gives the probability of any particular sequence with x successes and n-x failures

Mean of a Binomial Distribution and formula to calculate

The expected number of successes for a binomial random variable, calculated by the formula $E(X) = \mu = n p$.

Variance of a Binomial Distribution

The measure of dispersion for a binomial random variable, calculated using the formula $Var(X) = \sigma^2 = n p (1 - p)$.

Standard Deviation of a Binomial Distribution

The standard deviation of a binomial random variable, calculated using the formula $SD(X) = \sigma = \sqrt{n p (1 - p)}$.

How to find the binomial probabilities in excel

• =BINOM.DIST

The Excel function with syntax =BINOM.DIST(x, n, p, cumulative) where FALSE calculates the probability mass function $P(X = x)$ and TRUE calculates the cumulative probability function $P(X \le x)$.

Poisson Process

An experiment where the number of successes in a specified time or space interval can be any integer from zero to infinity, counts in nonoverlapping intervals are independent, and the probability of success is proportional to interval size.

EX:

• number of spam emails received in a month

• number of customers who use a new banking app in a day

• the number of defects in a 50-yard roll of fabric

Poisson Random Variable

The number of successes achieved within a specified time or space interval.

• such as defects in a 50-yard roll of fabric or customer app visits in a day.

Poisson Probability Distribution Formula

The formula used to compute the probability of $x$ successes in an interval: $P(X = x) = \frac{e^{-\mu} \mu^x}{x!}$ for $x = 0, 1, 2, \dots$, where $\mu$ is the mean number of successes and $e \approx 2.718$.

Summary Measures of a Poisson Random Variable

The mean and variance of a Poisson random variable both equal $\mu$ ($E(X) = \mu$, $Var(X) = \sigma^2 = \mu$), and the standard deviation equals the square root of the mean ($SD(X) = \sigma = \sqrt{\mu}$).

explain finding the Poisson distribution in excel

• POISSON.DIST

The Excel function with syntax =POISSON.DIST(x, \mu, cumulative) where FALSE evaluates the probability mass function $P(X = x)$ and TRUE evaluates the cumulative probability function $P(X \le x)$.