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# Exam 2 formula sheet STT 315

MSU

GPA 3.6

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## Popular in Intro to Business Statistics

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This 2 page Study Guide was uploaded by Jordyn on Friday April 1, 2016. The Study Guide belongs to STT 315 at Michigan State University taught by Eroglu in Fall 2015. Since its upload, it has received 27 views. For similar materials see Intro to Business Statistics in Business at Michigan State University.

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Date Created: 04/01/16

Random variable (rv) is a numerical variable whose value depends on chance—values are associated with outcomes of a random experiment where one and only one numerical value is assigned to each sample point Discrete—if it either takes finitely many or countable infinitely many values (its numbers can be enlisted in an infinite sequence) Continuous—if The set of possible values X can take is either an interval or a disjoint union of intervals on real line—if it can assume unaccountably many values o Example: total amount of rainfall Discrete random variable: P (x ≤ 1) Probability distribution function: P(x)= P(X=x) Expectation: μ = E(x) =Σ xp(x) P (x > 12) = 1 – P (x ≤ 12) Variance: σ = E[(x-μ) ] = Σ(x-μ) p(x) P (x < 3) = p (x ≤ 2) 2 P (x < 7) = P (x ≤ 7) – p (x = Standard deviation σ = +√σ For any real numbers a and b 7) E(aX ± b) = aE(X)±b P (x≥3) = 1 – P (x ≤2) P (2 ≤ x ≤ 4) = P(x ≤ 4) – σ(aX±b) = |a|σ(X) BINOMIAL Characteristics for binomial experiment: The experiment consists of n identical trials, There are only two possible outcomes on each trial – ‘success’ and ‘failure’, The probability of success remains the same from trial to trial, The trials are independent (n), The binomial random variable X is the number of successes in n trials, X assumes values 0,1,2… ,n. (Non-negative whole number) P(x)=( xp q n-x p(x)=P(X-x)= probability of x successes, n= number of trials, x= number of successes in n trials, (x=0,1,…,n), n-x=number of failures, p=P(sucess), and q=P(failure)=1-p binomcdf(n,p,x) mean: μ=np 2 variance: σ =npq standard deviation: σ = √npq POISSON Characteristics of Poisson random variable: The experiment consists of counting number of times an event occurs during a given unit of time or in a given area or volume (any unit of measurement), The probability that an event occurs in a given unit of time, area, or volume is the same for all units, The number of events that occur in one unit of time, area, or volume is independent of the number that occur in any other mutually exclusive unit, The mean number of events in each unit is denoted by λ, a positive real number -λ x P(x)=e *(λ /x!) p(x) = P(X=x) given λ, λ = mean (expected) number of events in unit, e = 2.71828 … (base of natural logarithm), x = number of events per unit Poissonpdf (λ,x) = poissonpdf (mean, x-value) μ = E(X) = λ σ = √λ the mean number of events in each unit is λ > 0 HYPERGEOMETRIC RANDOM VARIABLE The hypergeometric random variable X is the number of successes in the draw of n elements p(x) probability of x successes, N = total number of elements, r = number of successes in the N elements, n = number of elements drawn, x = number of successes in the n selections, where max {0, n – (N -r)} ≤ x ≤ min {r, n} μ = nr/N σ = r(N-r)n(N-n) 2 N (N-1) In calculator (r)= n nCr r Sampling with replacement vs sampling without replacement r elements are successes, and N − r are failures In with replacement sampling the selected objects are returned to the population before each draw—binomial distribution with n and p = r/N In without replacement sampling the selected objects are NOT returned to the population before each draw— hypergeometric distribution with parameters N, r, and n NORMAL DISTRIBUTION Can be used to approximate discrete probability distributions—empirical rule is based on normal distribution In many data-sets, the histograms are roughly symmetric, unimodal and bell-shaped, these distributions are usually modeled by normal distribution Location of normal distribution is determined by μ Dispersion of normal distribution is determined by σ Normalcdf (a, b, μ, σ)—a: lower bound, b: upper bound, μ: mean, σ: standard deviation To find the area : p invNorm (p, μ, σ)—area: enter the value of p, μ: mean, σ: standard deviation We want to find x 0invNorm is used when areas is given but want to find a number on the horizontal line Used to find area to left (x ≤ 6) ex: below what height—means the area to the left Standard normal distribution: μ=0 and σ=1, if μ and σ aren’t specified μ=0 and σ=1 so a= z and b1= z 2 Z score = x-μ p(x≤a): z = (a+0.5) – μ σ σ if given z score and asked to find the area to the left normalcdf(-9^99, z-score) Normal approximation of binomial distribution Use z-scores If n is very large (theoreticallyn∞), then the probability distribution of X is approximately normal with mean µ = np and standard deviation σ = √npq where q=1-p Continuity correction is required-- P(X≤x) ≈ P(Normal ≤x – 0.5), P(X=x) ≈ P(x-0.5 ≤ Normal ≤ x + 0.5) If the interval np±3√npq is contained in the interval (0,n) then n is “large enough” Are the data normally distributed Examine a normal probability plot for the data. If the data are approximately normal, the points will fall (approximately) on a straight line Uniform probability distribution Continuous random variables that appear to have equaly likely outcomes over their range of possible values posses a uniform probability distribution—looks like a box If x is a uniform rv in the interval (c,d) then p(a<x<b) = (b-a)/(d-c) If c ≤ a < b ≤ d: μ = (c+d)/2 σ = (d-c)/ √12 Exponential probability distribution function If x is an exponential rv with mean σ, then p(x>a) = e -a, a>0 –a/θ -b/θθ P(a<x<b) = e - e , 0 < a< b μ = θ, σ = θ

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