Takes numerical values that describe the outcomes of a random process.
Of a random variable, gives its possible values and their probabilities.
discrete random variable
χ takes a fixed set of possible values with gaps between them.
mean (expected value) of a discrete random variable
Its average value over many, many trials of the same random process.
standard deviation of a discrete random variable
Measures how much the values of the variable typically vary from the mean in many, many trials of the random process.
The weighted average of squared deviations.
continuous random variable
Can take any value in an interval on the number line.
independent random variables
If knowing the value of χ does not help us predict the value of γ, then χ and γ are "independent random variables". In other words, two random variables are independent if knowing the value of one variable does not change the probability distribution of the other variable.
Arises when we perform 𝑛 independent trials of the same random process and count the number of times that a particular outcome (called a “success”) occurs. The four conditions for a binomial setting are: i) Binary? The possible outcomes of each trial can be classified as “success” or “failure.”; ii) Independent? Trials must be independent. That is, knowing the outcome of one trial must not tell us anything about the outcome of any other trial.; iii) Number? The number of trials 𝑛 of the random process must be fixed in advance.; iv) Same probability? There is the same probability of success p on each trial.
binomial random variable
The count of successes χ in a binomial setting. The possible values of χ are 0, 1, 2, …, n.
The probability distribution of χ. Any "binomial distribution" is completely specified by two numbers: the number of trials 𝑛 of the random process and the probability p of success on each trial.
The count of the number of arrangements of x successes in 𝑛 trials.
When taking a random sample of size 𝑛 from a population of size 𝑁, we can treat individual observations as independent when performing calculations as long as 𝑛 < 0.10𝑁.
Large Counts condition
Suppose that a count χ of successes has the binomial distribution with 𝑛 trials and success probability 𝑝. The "Large Counts condition" says that the probability distribution of χ is approximately Normal if 𝑛𝑝≥10 and 𝑛(1−𝑝)≥10. That is, the expected numbers (counts) of successes and failures are both at least 10.
Arises when we perform independent trials of the same random process and record the number of trials it takes to get one success. On each trial, the probability p of success must be the same.
geometric random variable
The number of trials χ that it takes to get a success in a geometric setting.
The probability distribution of χ is a "geometric distribution" with probability 𝑝 of success on any trial. The possible values of χ are 1, 2, 3, . . . .