Random Variables Unit Vocabulary

12th Grade

random variable

variable that takes numerical values that describe the outcomes of a random process

probability distribution

gives the possible values of a random variable and their probabilities

discrete random variable

takes a set of possible values with gaps between them on a number line

mean of a discrete random variable; expected value

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 same random process

binomial setting (fill in the blanks)

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.

Four conditions for a binomial setting are:
Binary? The possible outcomes of each trial can be classified as “success” or “failure”
Independent? Trials are independent. That is, knowing the outcome of one trial must not tell us anything about the outcome of any other trial.
Number? The number of trials n of the random process is fixed in advance.
Same probability? There is the same probability of success p on each trial.

binomial random variable

the count of successes X in a binomial setting. The possible values of X are 0, 1, 2, …, n.

binomial distribution

The probability distribution of X

binomial coefficient

The number of ways to arrange x successes among n trials given by binomial coefficient

binomial probability

suppose that X is a binomial random variable with n trials and probability p of successes on each trial. The probability of getting exactly x successes in n trials (x = 0, 1, 2, …, n) is (picture)

mean of binomial random variable

If a count X of successes has a binomial distribution with number of trials n and probability of success p, the mean of X is (picture)

standard deviation of a binomial random variable

If a count X of success has a binomial distribution with number of trials n and probability of success p, the standard deviation of X is (picture)

10% Condition

When selecting a random sample of size n without replacement from a population of size N, we can treat individual observations as independent when performing calculations as long as n <= 10%N

continuous random variable

can take any value in a specified interval on the number line

density curve

models the probability distribution of a continuous random variable with a curve that:

• is always on or above the horizontal axis

• has an area of exactly 1 underneath it

The area under the density curve and above any specified interval of values on the horizontal axis gives the probability that the random variable falls in that interval.

normal distribution

distribution described by a normal curve. Any normal distribution is completely specified by two parameters, its mean and standard deviation. The mean is located at the balance point of the symmetric normal distribution and is the same as the median. The standard deviation is the distance from the mean to the change-of-curvature points on either side.

normal curve

important kind of curve that is symmetric, single-peaked, and bell-shaped

mean

standard deviation

normal random variable

a continuous random variable whose probability distribution is described by a normal density curve

Empirical Rule

• About 68% of the values falls within 1 standard deviation of the mean

• About 95% of the values fall within 2 standard deviation of the mean

• 99.7% of the values fall within 3 standard deviation of the mean

standard normal distribution

the normal distribution with mean 0 and standard deviation 1

point estimator

a statistic that provides an estimate of a population parameter

point estimate

the value of a point estimator from a sample

sampling variability

refers to the fact that different random samples of the same size from the same population produce different values for a statistic

sampling distribution

the distribution of values taken by the statistic in all possible samples of the same size from the same population

unbiased estimator

a statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the value of the parameter being estimatedbi