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Chapter 21 Random Variables Discrete: Bernoulli, Binomial, Geometric, Poisson Continuous: Uniform, Exponential, Gamma, Normal Expectation & Variance, Joint Distributions, Moment Generating Functions, Limit Theorems
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Chapter 22 Definition of random variable A random variable is a function that assigns a number to each outcome in a sample space. If the set of all possible values of a random variable X is countable, then X is discrete. The distribution of X is described by a probability mass function: Otherwise, X is a continuous random variable if there is a nonnegative function f(x), defined for all real numbers x, such that for any set B, f(x) is called the probability density function of X.
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Chapter 23 pmf’s and cdf’s The probability mass function (pmf) for a discrete random variable is positive for at most a countable number of values of X: x 1, x 2, …, and The cumulative distribution function (cdf) for any random variable X is F(x) is a nondecreasing function with For a discrete random variable X,
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Chapter 24 Bernoulli Random Variable An experiment has two possible outcomes, called “success” and “failure”: sometimes called a Bernoulli trial The probability of success is p X = 1 if success occurs, X = 0 if failure occurs Then p(0) = P{X = 0} = 1 – p and p(1) = P{X = 1} = p X is a Bernoulli random variable with parameter p.
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Chapter 25 Binomial Random Variable A sequence of n independent Bernoulli trials are performed, where the probability of success on each trial is p X is the number of successes Then for i = 0, 1, …, n, where X is a binomial random variable with parameters n and p.
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Chapter 26 Geometric Random Variable A sequence of independent Bernoulli trials is performed with p = P(success) X is the number of trials until (including) the first success. Then X may equal 1, 2, … and X is named after the geometric series: Use this to verify that
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Chapter 27 Poisson Random Variable X is a Poisson random variable with parameter > 0 if note: X can represent the number of “rare events” that occur during an interval of specified length A Poisson random variable can also approximate a binomial random variable with large n and small p if = np: split the interval into n subintervals, and label the occurrence of an event during a subinterval as “success”.
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Chapter 28 Continuous random variables A probability density function (pdf) must satisfy: The cdf is: means that f(a) measures how likely X is to be near a.
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Chapter 29 Uniform random variable X is uniformly distributed over an interval (a, b) if its pdf is Then its cdf is: all we know about X is that it takes a value between a and b
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Chapter 210 Exponential random variable X has an exponential distribution with parameter > 0 if its pdf is Then its cdf is: This distribution has very special characteristics that we will use often!
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Chapter 211 Gamma random variable X has an gamma distribution with parameters > 0 and > 0 if its pdf is It gets its name from the gamma function If is an integer, then
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Chapter 212 Normal random variable X has a normal distribution with parameters and if its pdf is This is the classic “bell-shaped” distribution widely used in statistics. It has the useful characteristic that a linear function Y = aX+b is normally distributed with parameters a b and (a . In particular, Z = (X – )/ has the standard normal distribution with parameters 0 and 1.
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Chapter 213 Expectation Expected value (mean) of a random variable is Also called first moment – like moment of inertia of the probability distribution If the experiment is repeated and random variable observed many times, it represents the long run average value of the r.v.
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Chapter 214 Expectations of Discrete Random Variables Bernoulli: E[X] = 1(p) + 0(1-p) = p Binomial: E[X] = np Geometric: E[X] = 1/p (by a trick, see text) Poisson: E[X] = the parameter is the expected or average number of “rare events” per interval; the random variable is the number of events in a particular interval chosen at random
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Chapter 215 Expectations of Continuous Random Variables Uniform: E[X] = (a + b)/2 Exponential: E[X] = 1/ Gamma: E[X] = Normal: E[X] = the first parameter is the expected value: note that its density is symmetric about x = :
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Chapter 216 Expectation of a function of a r.v. First way: If X is a r.v., then Y = g(X) is a r.v.. Find the distribution of Y, then find Second way: If X is a random variable, then for any real- valued function g, If g(X) is a linear function of X:
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Chapter 217 Higher-order moments The nth moment of X is E[X n ]: The variance is It is sometimes easier to calculate as
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Chapter 218 Variances of Discrete Random Variables Bernoulli: E[X 2 ] = 1(p) + 0(1-p) = p; Var(X) = p – p 2 = p(1-p) Binomial: Var(X) = np(1-p) Geometric: Var(X) = 1/p 2 (similar trick as for E[X]) Poisson: Var(X) = the parameter is also the variance of the number of “rare events” per interval!
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Chapter 219 Variances of Continuous Random Variables Uniform: Var(X) = (b - a) 2 /2 Exponential: Var(X) = 1/ Gamma: Var(X) = 2 Normal: Var(X) = 2 the second parameter is the variance
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Chapter 220 Jointly Distributed Random Variables See text pages 46-47 for definitions of joint cdf, pmf, pdf, marginal distributions. Main results that we will use: especially useful with indicator r.v.’s: I A = 1 if A occurs, 0 otherwise
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Chapter 221 Independent Random Variables X and Y are independent if This implies that: Also, if X and Y are independent, then for any functions h and g,
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Chapter 222 Covariance The covariance of X and Y is: If X and Y are independent then Cov(X,Y) = 0. Properties:
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Chapter 223 Variance of a sum of r.v.’s If X 1, X 2, …, X n are independent, then
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Chapter 224 Moment generating function The moment generating function of a r.v. X is Its name comes from the fact that Also, if X and Y are independent, then And, there is a one-to-one correspondence between the m.g.f. and the distribution function of a r.v. – this helps to identify distributions with the reproductive property
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