AP Statistics Chapter 6: Random Variables
What is the effect on a random variable of adding or subtracting by a constant a?
- adds (a) to measures of center and location (mean, median, quartiles, percentiles) -does not change the shape or measures of spread (range, IQR, standard deviation)
What is the effect on a random variable of multiplying or dividing by a constant b?
-multiplies/divides measures of center and location (mean, median, quartiles, percentiles) by b -multiplies/divides measures of spread (range, IQR, standard deviation) by |b| -does not change the shape of the distribution
A binomial setting
A binomial setting arises when we perform several independent trials of the same chance process and record the number of times a particular outcome occurs.
Continuous Random Variable
A continuous random variable x takes all values in an interval of numbers. The probability distribution of x is described by a density curve. The probability of any event is the area under the density curve and above the values of x that make up the event.
What is the difference between a discrete random variable and a continuous random variable?
A discrete random variable has a set number of possibilities whereas a continuous random variable has an infinite amount of possibilities.
Geometric Setting
A geometric setting arises when we perform independent trials of the same chance process and record the number of trials until a particular outcome occurs.
Discrete Random Variable
A random variable having probability assigned.
Random Variable
A random variable takes numerical values that describe the outcomes of some chance process.
What is the mean of a discrete random variable x?
An average of the possible values of x, but with an important change to take into account the fact that not all outcomes may be equally likely.
What happens if two independent normal random variables are combined?
Any sum or difference or independent normal random variables is also normally distributed.
Conditions of a Binomial Setting
BINS Binary- the possible outcomes of each trial can be classifies as "success" or "failure" Independent- trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. Number- the number of trials n of the chance process must be fixed in advance Success- on each trial, the probability p of success must be the same
Conditions of a Geometric Setting
BITS Binary-the possible outcomes of each trial can be classifies as "success" or "failure" Independent- trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. Trials-the goal is to count the number of trials until the first success occurs Success- on each trial, the probability p of success must be the same
What are the two requirements for the probability distributions of discrete random variables?
Every probability is a number between 0 and 1, and the sum of the probabilities is 1.
Geometric probability formula
If Y has the geometric distribution with probability p of success on each trial, the possible values of Y are 1,2,3,.... If k is any one of these values, P(Y=k)=(1-p)^(k-1) (p)
The mean/expected value of a Geometric random variable
If Y is a geometric random variable with the probability of success p on each trial, then its mean (expected value) is E(Y)=µ(subscript y)=(1/p). That is, the expected number of trials required to get the first success is 1/p.
What are independent random variables?
If knowing whether any event involving x alone has occurred tells us nothing about the occurrence of any event involving y alone, and vise versa, then x and y are independent random variables.
Binomial Probabilities
If x has the binomial distribution with n trials and probability p of success on each trial, the possible values of x are 0,1,2,...,n. If k is any one of these given values,
The mean of the difference of random variables
In general, the mean of the difference of several random variables is the difference of their means.
Define the mean of the sum of random variables
In general, the mean of the sum of several random variables is the sum of their means.
How does multiplying by a constant effect the variance?
Multiplying a random variable by a constant b multiplies the varience by b²
Does the expected value of a random value have to equal on the the possible values of the random variable?
No, because it is a long run average.
Normal Approximation for Binomial Distributions
Suppose that a count x has the binomial distribution with n trials and success probability p. When m is large, the distribution of x is approximately Normal with mean: µₓ=np and standard deviation: σₓ=√(np(1-p)) As a rule of thumb, we will use the Normal approximation when n is so large that np≥10 and (np(1-p))≥10 That is, the expected number of success and failures are both at least 10
Binomial random variable
The count x of successes in a binomial setting is a binomial random variable
What is the expected value?
The expected value is the mean and has the notation E(x)
In a probability histogram, what does the height of each bar represent?
The height represents the probability of the outcome.
Geometric random variable
The number of trials Y that it takes to get a success in a geometric setting is a geometric random variable. The possible values of Y are 1,2,3,...
Binomial coefficient
The number of ways arranging k successes among n observations is given by the binomial coefficient, which is NOT related to fractions(!!!)
If x is a continuous random variable, how is the probability distribution of x described?
The probability distribution for a continuous random variable assigns probability to intervals of outcomes rather than to individual outcomes.
Parameters of a geometric setting
The probability distribution of Y is a geometric distribution with parameter p, the probability of a success on any trial.
Probability Distribution
The probability distribution of a random variable gives its possible values and their probabilities.
Binomial Distribution and its Parameters
The probability distribution of x is a binomial distribution with parameters n and p, where n is the number of trials of the chance process and p is the probability of a success on any one trial. The possible values of x are whole numbers form 0 to n.
In a probability histogram, what is the sum of the height of each bar?
The sum adds up to 1 because the heights represent probability.
Define linear transformation
The transformation when adding/subtracting, or multiplying/dividing by a constant.
Sampling without replacement condition
When taking an SRS of size n from a population of size N, we can use a binomial distribution to model the count of success in the sample as long as n≤(1/2)N
When do you add the variances of two random variables?
When the two random variables are independent.
Using a calculator for binomial probabilities
binompdf(n, p, k) computes P(x=k) binomcdf(n, p, k) computes P(x≤k)
How to calculate the mean and standard deviation of a Binomial random variable
μₓ=np σₓ=√(np(1-p)) where n=number of trials, and p= probability of success
