Statistics Chapter 6
Variance of the Sum of Independent Random Variables
For any two independent random variables X and Y, if T=X+Y, then the variance of T is σT2=σx2+σy2 In general, the variance of the sum of several independent random variables is the sum of their variances.
Geometric Probability
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−1p
Effects of a Linear Transformation on the Mean and Standard Deviation
If Y=a+bX is a linear transformation of the random variable X, then ● the probability distribution of Y has the same shape as the probability distribution of X. ● μy=|b|μx ● σy= |b|σx(since b could be a negative number).
Mean and Standard Deviation of a Binomial Random Variable
If a count X has the binomial distribution with number of trials n and probability of success p, the mean and standard deviation of X are μx=np σx=√np(1−p)
Mean, Standard Deviation of X
Mean and Standard Deviation of a Binomial Random Variable If a count X has the binomial distribution with number of trials n and probability of success p, the _____ and _____ are μx=np σx=√np(1−p)
Effect on a Random Variable of Multiplying (or Dividing) by a Constant
Multiplying (or dividing) each value of a random variable by a number 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.
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 in the sample as long as n≤1/10N
Mean of the Sum of Random Variables
For any two random variables X and Y, if T=X+Y, then the expected value of T is E(T)=μt=μx+μy In general, the mean of the sum of several random variables is the sum of their means.
Mean (Expected Value) of a Discrete Random Variable
Suppose that X is a discrete random variable whose probability distribution is __________________________ Value: x1 x2 x3 ... Probability: p1 p2 p3 ... ___________________________ To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: 𝜇𝑋=𝐸(𝑋)=𝑥1𝑝1+𝑥2𝑝2+𝑥3𝑝3+⋯ 𝜇𝑋=Σ𝑥𝑖𝑝𝑖
Variance and Standard Deviation of Discrete Random Variable
Suppose that X is a discrete random variable whose probability distribution is __________________________ Value: x1 x2 x3 ... Probability: p1 p2 p3 ... ___________________________ and that 𝜇𝑋 is the mean of X. The variance is 𝑉𝑎𝑟(𝑋)=𝜎𝑋2=(𝑋1− 𝜇𝑋)2𝑝1+(𝑋2− 𝜇𝑋)2𝑝2+(𝑋3− 𝜇𝑋)2𝑝3+⋯ 𝑉𝑎𝑟(𝑋)=𝜎𝑋2=Σ(𝑋𝑖− 𝜇𝑋)2𝑝𝑖
Independent Random Variables
If knowing whether any event involving x along has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are _____.
Binomial Probability
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 P(X=k)=(n,k)pk(1−p)n−k With our formula in hand, we can now calculate any binomial probability.
Normal Approximation for Binomial Distributions
Suppose that a count X has the binomial distribution with n trials and success probability p. When n 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 n(1−p)≥10 That is, the expected number of successes and failures are both at least 10.
Geometric Setting, Binary, Independent, Trials, Successes
A _____ arises when we perform independent trials of the same chance process and record the number of trials until a particular outcome occurs. The four conditions for a geometric setting are: -_____ - The possible outcomes of each trial can be classified as "success" or "failure" -_____ - Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. -_____ - The goal is to count the number of trials until the first success occurs. -_____ - On each trial, the probability p of success must be the same.
Binomial Setting
A _____ 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: -Binary - The possible outcomes of each trial can be classified 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.
Continuous Random Variable
A _____ 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.
Random Variable
A _____ takes numerical values that describe the outcomes of some chance process.
Discrete Random Variables and Their Probability Distributions
A discrete random variable X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xiand their probabilities pi: __________________________ Value: x1,x2,x3,... Probability: p1,p2,p3,... ___________________________ The probabilities pimust satisfy two requirements 1. Every probability piis a number between 0 and 1. 2. The sum of the probabilities is 1: p1+ p2+ p3=1.. To find the probability of any event, add the probabilities piof the particular values xithat make up the event.
Effect on a Random Variable of Adding (or Subtracting) a Constant
Adding the same number a (which could be negative) to each value of a random variable: ● Adds a to measures of center and location (mean, median, quartiles, percentiles). ● Does not change shape or measures of spread (range, IQR, standard deviation).
Discrete Random Variable (X)
Discrete Random Variables and Their Probability Distributions A _____ takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values xiand their probabilities pi: __________________________ Value: x1,x2,x3,... Probability: p1,p2,p3,... ___________________________ The probabilities pimust satisfy two requirements 1. Every probability piis a number between 0 and 1. 2. The sum of the probabilities is 1: p1+ p2+ p3=1.. To find the probability of any event, add the probabilities piof the particular values xithat make up the event.
Variance of the Difference of Random Variables
For any two independent random variables X and Y, if D=X-Y, then the variance of D is σD2=σx2-σy2
Mean of the Difference of Random Variables
For any two random variables X and Y, if D=X-Y, then the expected value of D is E(D)=μD=μx-μy In general, the mean of the difference of several random variables is the difference of their means.
Standard Deviation
The _____ of X, 𝜎𝑋, is the square root of the variance.
Probability Distribution
The _____ of a random variable gives its possible values and their probabilities.
Binomial Random Variable
The count X of successes in a binomial setting is a _____.
Geometric Random Variable
The number of trials Y that it takes to get a success in a geometric setting is a _____.
Binomial Distribution, Binomial Coefficient
The probability distribution of X is a _____ 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 the whole numbers from 0 to n. The number of ways of arranging k successes among n observations is given by the _____ (𝑛/𝑘)=𝑛!𝑘!/(𝑛−𝑘)! for 𝑘=0,1,2,...,𝑛 𝑤ℎ𝑒𝑟𝑒 𝑛!=𝑛(𝑛−1)(𝑛−2)....(3)(2)(1) and 0!=1.
Geometric Distribution
The probability distribution of Y is a _____ with parameter p, the probability of a success on any trial. The possible values of Y are 1,2,3...
Variance
Variance and Standard Deviation of Discrete Random Variables Suppose that X is a discrete random variable whose probability distribution is __________________________ Value: x1 x2 x3 ... Probability: p1 p2 p3 ... ___________________________ and that 𝜇𝑋 is the mean of X. The _____ is 𝑉𝑎𝑟(𝑋)=𝜎𝑋2=(𝑋1− 𝜇𝑋)2𝑝1+(𝑋2− 𝜇𝑋)2𝑝2+(𝑋3− 𝜇𝑋)2𝑝3+⋯ 𝑉𝑎𝑟(𝑋)=𝜎𝑋2=Σ(𝑋𝑖− 𝜇𝑋)2𝑝𝑖
