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### stat400lec9

Course: STAT 400, Fall 2008
School: University of Illinois,...
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400 Statistics Lecture 9 Section 2.4 Bernoulli Trials Bernoulli Experiment: A Bernoulli experiment is a random experiment, the outcome of which can be classified as success and failure. Let X be random variable of Bernoulli distribution. Outcome Probability Success 1 p Failure 0 q=1-p Notation: X ~ Ber (p) pmf Mean E[X]=p Variance Var[X]=pq A sequence of Bernoulli Trials: A sequence of Bernoulli experiments are...

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400 Statistics Lecture 9 Section 2.4 Bernoulli Trials Bernoulli Experiment: A Bernoulli experiment is a random experiment, the outcome of which can be classified as success and failure. Let X be random variable of Bernoulli distribution. Outcome Probability Success 1 p Failure 0 q=1-p Notation: X ~ Ber (p) pmf Mean E[X]=p Variance Var[X]=pq A sequence of Bernoulli Trials: A sequence of Bernoulli experiments are conducted independently and the probability of success (p) remains the same. Ping Ma Lecture 9 Fall 2006 -1- Example: For a million of instant lottery tickets, suppose that 20% are winners, If 5 tickets are purchased, (1) what is the probability we have (0,1,0,1,0)? (2) what is the probability we have (0,0,1,1,0)? (3) What is the probability we have two winning tickets? Binomial distribution 1. A Bernoulli experiment is performed n times 2. The trials are independent 3. The probability of success on each trial is a constant p; The probability of failure each on trial is a constant 1-p 4. The random variable X equals the number of successes in the n trials. X~Bin(n,p) ( b(n,p) ) n and p are parameters of binomial distribution. Ping Ma Lecture 9 Fall 2006 -2- Example: What is the probability that we have at most two winning tickets among 5 tickets? Cumulative distribution function (CDF) F(x)=P(X x) Example: Your job is to examine light bulbs on an assembly line. You count the number of defective light bulbs after examining 10 light bulbs. Let X = number of defective light bulbs P (defective) = .2 N = 10 1. What is the probabil...

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University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 10 Section 2.5 The Moment-generating function Moment-generating function (Laplace Transform)tx M(t)=E[etX]= e f (x )M'(t)= M&quot;(t)= xetxf (x ) f ( x)M'(0)= M&quot;(0)= x f ( x ) =E[X] x2x e2 txf ( x ) =E[X2]Thus
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 11 Section 2.6 The Poisson DistributionPoisson distributionf ( x) =x e - , x=0,1,2,. x!E[X] = Example:Var[X]= (1) Let X have a Poisson distribution with a variance of 3, Find P(X=2), P(X 6)(2) If X has a Poisson
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 12 Continuous Type Data Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Note: When a random variable X describes a random phenomenon, the sample space S just li
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 13 Continuous Distributions Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Note: When a random variable X describes a random phenomenon, the sample space S jus
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 14 Waiting Time Review: Continuous distribution Percentile : The (100p)th percentile p is a number such that the area under f(x) to the left of p is p. That is:first quartile: 25th percentile median: 50th percentile third qua
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 15 Waiting Time Review: Example: Customers arrive at Staples according a Poisson distribution with mean rate 1/3 per minute. On Thanksgiving morning, Staples opens 5:00am in the Morning. The first 10 customers will get a free f
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 16 Review Random Variable: (R.V.) is a variable whose value is a numerical outcome of a random phenomenon. Notation: X, Y, Z Probability Theorem 2.1-1: Complementary Rule: P (A') = 1-P (A) Theorem 2.1-2: P( )=0 Theorem 2.1-3:
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 17 Functions of a Random Variable X is a continuous random variable Y is a continuous random variable defined as Y=u(X). What is the distribution of Y CDF of Y is G(y)=P(Y&lt;y)=P[u(X)&lt;y] pdf of Y is g(y)=G'(y) Example: X has a ga
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Correlation CoefficientDiscrete random variables (X,Y)Joint probability mass function of (X,Y) x1 y1 y2 y3 f(x1, y1) f(x1, y2) f(x1, y3) . . ym f(x1, ym) x2 f(x2, y1) f(x2, y2) f(x2, y3) . f(x2, ym) x3 f(x3, y1) f(x3, y2) f(x3, y3)
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Conditional DistributionDiscrete random variables (X,Y)Joint probability mass function of (X,Y) x1 y1 y2 y3 f(x1, y1) f(x1, y2) f(x1, y3) . . ym f(x1, ym) x2 f(x2, y1) f(x2, y2) f(x2, y3) . f(x2, ym) x3 f(x3, y1) f(x3, y2) f(x3, y3)
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Transformation of Random Variables Too complicated to discuss in this class One important distribution to remember Let U and V are two independent chi-square random variables with r1 and r2 degrees of freedom, respectively.F= U / r1 V
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 4.5 Random Sample If the distributions of the independent random variables X1 and X2 are the same, the collection of the two random variables X1 and X2 is called a random sample of size 2 from the distribution.If the distribu
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 4.6 Distribution of sums of the independent random variables Theorem 4.6-1 If X1, X2, ., Xn are n independent random variables with respective 2 2 2 means 1, 2, ., n and variances 1 , 2 , ., n , then the mean and n varianc
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 The Normal Distribution Normal Curves: Describe the Normal Distribution All normal curves have the same shape: Probability density function1 ( x - )2 f ( x) = exp[ - ] 2 2 2Graph: bell shaped-&lt; x &lt; Mean: : at the center of
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.3 Distribution of sums of the independent random variables Review If X1, X2, ., Xn are n independent random variables with respective 2 2 2 means 1, 2, ., n and variances 1 , 2 , ., n , then the mean and n variance of Y=
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.4 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.5 Central limit theorem Central Limit Theorem (CLT) If X1, X2, ., Xn are observations of a random sample of size n from a distribution with mean and variance 2 , Then we have W=X - X i - n = N(0,1) as / n nnIn anoth
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 5.6 Bivariate normal distribution Two random variables X and Y have a bivariate normal distribution with 2 means X and Y , variances X and Y2 , correlation coefficient .2 X X X ~ , N Y Y X Y X Y
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Lecture 29 Review Continuous distribution: Probability density function Properties of p.d.f f(x): (a) f(x)&gt;0; (b) f ( x )dx = 1 ; (c) P(a&lt;X&lt;b)= a f ( x )dx Cumulative distribution function (c.d.f) F(x)= P( X x ) = F'(x)=f(x) Expected
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Chapter 6 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance i
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.2 Method of momentsReview:We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Confidence intervals for means The Statistical End Goal: draw conclusions about the data we have collected and analyzed. Every time we estimate using statistic X , we will never get the same answer. Due to sampling variability Need t
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.5 Confidence Intervals for Proportions Binomial distribution Let Y ~ b(n,p) Objective: construct confidence interval for pY - np Y /n- pW= np(1 - p ) = p(1 - p ) / n N(0,1) &quot;sufficiently large&quot;: np 5 and n(1-p) 5Ping
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Section 6.7 Sample Size Calculation Review: Confidence Interval Form: estimate margin of error n s x t / 2 * nx z / 2 *Y Y / n (1 - Y / n ) z / 2 n nThe length of the interval = 2*margin of error Tow factors associated with w
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypothesis Confidence Intervals: Creates an interval where we think the true parameter we are estimating will fall with a certain probability or level of confidence. Serve the purpose when the goal is to estimate
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests of Statistical Hypotheses Simple hypothesis Composite HypothesisTypes of Error Type I Error: If we reject Ho when Ho is true Type II Error: If we fail to reject Ho when Ho is false Ho True Reject Ho Type I error Ha True Correct
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Tests about Proportions Recall:^ Proportions: ( p = ) Y nCount the number of successes and take into account the sample Normal Approximation of proportions Draw a random sample of size n from a large population with p = P ^ (Success
University of Illinois, Urbana Champaign - STAT - 400
Statistics 400 Comparing Two Means and Two Variances Two Sample Problems: Compare the responses in two groups Each group is an individual sample from a distinct population The responses from either group are independent of each other When a two-sa
University of Illinois, Urbana Champaign - STAT - 400
Final Review We consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Estimator: The function of X1, X2,., Xn used to estimate , say the st
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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UNC Wilmington - CHM - 101
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Uni. Worcester - CS - 4341
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