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AP STATS Chapter 6
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Terms in this set (22)
Discrete Random Variable
X takes a fixed set of possible values with gaps between. (Only certain Numbers)
What requirement does pi require?
The Probabilities are between 1 and 0.
All of the Probabilities add up to 1
Expected Value (Mean) Formula
Mean of X equals the sum of Every X times its own probability.
Mew of X= E(x)= ∑xi Pi
The Variance of a Discrete Random Variable
Var(x) = Standard deviation ² = (x1-Mean of X)²p1 +.....
Continuous Random Variable
Takes on all values in an interval of numbers. Use a Probability Distribution Graph!
Transforming Data with Adding/ Subtracting a Constant to every data value
Increases/Decreases the measures of center and location by that number
Does not affect the Standard Deviation or measures of spread (range, IQR)
Transforming Data with Multiplying/ Dividing a Constant to every data value
Multiplies/Divides affects measures of Center, Location, and Spread (mean, standard deviation, median, mode, percentiles, etc.)
Does Not Change Shape
Linear Transformation Formula
The New Mean (Y) equals a + b times mean of X
The New Standard Deviation (Y) equals the absolute value of B times the mean of X
Independent Random Variables
If Knowing any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are Independent Random Variables.
For any 2 random variables X and Y, if D= X-y, then the variance of D is
Variance X plus variance Y.
Binomial Setting
A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs, the four conditions for a binomial setting are BINS
BINS
1)Binary? The Possible outcomes of each trial can be classified as "success" or "failure"
2)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.
3)Number? The number of trials n of the chance process must be fixed in advance.
4) Success? On each trial, the probability p of success must be the same.
Binomial Distribution
The Count of X success in a binomial setting is a binomial random variable. The probability distribution of X is a binomial distribution w/ 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 the whole numbers from 0 to n
Binomial Coefficient
The number of ways of arranging k successes among n observations is given by the binomial coefficient
(n)
( )= n Factorial ÷ k Factorial (n-k)Factorial
(k)
Just use 2nd Vars A on ti-84 calculator
How to find the mean and standard deviation of a binomial random variable
Mean of x= np
Standard deviation of x= √Mean of x (1-p)
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 successes of the sample as long as
n ≤ .1 N
n(1-p) ≤ 10% of the population
Normal Approximation for Binomial Distributions
Use the Normal Approximation when n is so large that np ≥ 10
and n(1-p) ≥ 10. That is, the number of successes and failures are both at least 10.
Geometric Settings
A geometric setting arises when we perform independent trials of the same chance process and record the number of trials until a a particular outcome occurs. The four conditions for a geometric setting are BITS
BITS
1)Binary? The Possible outcomes of each trial can be classified as "success" or "failure"
2)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.
3)Trials? The goal is to count the number of trials until the first success occurs.
4) Success? On each trial, the probability p of success must be the same.
Geometric Random Variable
The number of trials Y that it takes to succeed in a geometric setting is a geometric random variable. The probability distribution of Y is a geometric distribution with parameter p, the probability of a success on any trial. The possible values of Y are 1,2,3, ....
Geometric Probability
If Y has the geometric distribution with each probability "p" of success on each trial, the possible values of U are 1,2,3, ... If k is any one of these values,
P(Y=k)= (1-p)^k-1 times p
Use 2nd Vars E
Mean (expected Value) of Y
Mean of Y = 1/p
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