Discrete Random Variables
A random variable is considered to be discrete if it can only map to a finite or countably infinite number of distinct values.
| Distribution | Parameter(s) | PMF \(P(Y=y)\) |
|---|---|---|
| Bernoulli | \(p\) | \(p^y(1-p)^{1-y}\) |
| Binomial | \(n\) and \(p\) | \((^n_y)p^y(1-p)^{n-y}\) |
| Geometric | \(p\) | \((1-p)^{y-1}p\) |
| Negative Binomial | \(r\) and \(p\) | \((^{y-1}_{r-1})p^{r-1}(1-p)^{y-r}\) |
| Poisson | \(\lambda\) | \(\frac{\lambda^y}{y!} e^{-\lambda}\) |
The probability mass function of discrete variable can be represented by a formula, table, or a graph. The Probability of a random variable Y can be expressed as \(P(Y=y)\) for all values of \(y\).
The cumulative distribution function provides the \(P(Y\leq y)\) for a random variable \(Y\).
A Bernoulli distribution is probability of having a success out of two outcomes. In essence, a coin flip with probability of success \(p\).
For a given sample, the number of success is represent with a binomial distribution. Binary outcomes are represented with either a Bernoulli of binomial distribution. For a binomial experiment, the following must be satisfied:
There is a fixed \(n\) trials
The there are two outcomes for each trial
The probability of success (\(p\)) is constant for each trial
The trials are independent of each other
The probability of a planted seed to germinate is 0.3. An arborist decides to plant 8 seeds in the community. What is the probability that 3 to 5 seeds will germinate?
The probability of a planted seed to germinate is 0.6. An arborist decides to plant 8 seeds in the community. What is the probability that at least 1 seed germinates?
The probability of a planted seed to germinate is 0.8. An arborist decides to plant 6 seeds in the community. What is the probability that an even number of seed will germinate?
The poisson distribution describes an experiment that measures that occurrence of an event at specific point and/or time period.
The number of industrial accidents at a particular manufacturing plant is found to be an average of 1 accident per month. In the previous month, there were 3 accidents. What is the probability of observing 3 or more accidents in a given month?
The number of industrial accidents at a particular manufacturing plant is found to be an average of 8 accidents per year. What is the probability of observing 15 or more accidents in a given year?
A geometric distribution models the number of Bernoulli trials until the first success (or failure).
Suppose 30% of the applicants for a certain industrial job possess the necessary skills. What is the probability that the first applicant with the necessary skills is found on the 5th interview.
A negative binomial distribution models the number of successful Bernoulli trials until the \(r\)th failure.
Ten percent of engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that third nondefective engine is found on the 4th trial.