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Math 310 – Probability and Statistics Final Exam Study Guide P(A B)=P(A) + P(B) – P(A B) Multiplication Principle: if your wardrobe consists of 5 pants, 4 shirts, and 3 sweaters, you have 60 (5*4*3) outfits. For a Permutation order matters - nPr, where n is the total number of items and r is the number that have to be arranged. nPr = n!/(n-r)! ex. If there are 10 people and 5 seats, write 10P5 P(A B C)=P(A) + P(B) + P(C) – P(A B) - P(B C) - P(A C) + P(A B C) For a Combination order doesn’t matters – nCr, where n is the total number of items and r is the number that are chosen (the number of ordered subsets of r objects from n objects). nCr = n!/(n-r)!r! ex. Probability of getting a poker hand of all spades, 13C5/52C5 P(E|F) is the Conditional Probability of E given F P(E|F)=P(EF)\P(F)\ P(AB)=P(A|B)P(B) P(BA)=P(B|A)P(A) Events A and B are Mutually Independent if and only if P(AB)=P(A)P(B), otherwise A and B are Dependent. P(AB)=P(A)+P(B)-P(AB) Bayes’ Formula: P(W)=P(B1W)+P(B2W)+P(B3W) Then if we know that a white chip was chosen and we want to see the probability that the white chip came from bowl 1 use this formula: P(B1|W)=P(B1W)/P(W) Bernoulli f(x)=px(1-p)1-x where xRange, =p, s2=p(1-p) Hypergeometric Function (N1Cx)(N2Cn-x)/(NCn), where N=# of items, N1=# of defective (or tagged, etc), N2=# left over, n=sample size =expected value (or mean)=E(x) s2=variance=E((x-)2) or E(x2)-2 =standard deviation=s2 Binomial use algebraic manipulation to change the values to search for in the table if it is not the proper format For a sample data =(1/n)(every value*the # of times it occurs), s2=(value-)2*occurrence of value this is done for every value When a problem is started it must be stated what x is (x is the # of telephone calls in an hour) and what x is declared as (x is a Poisson Dist.) Type of distribution and how it is defined: 1. Bernoulli – there are only 2 outcomes, success or failure. 2. Binomial - is when there is no replacement when choosing items from the sample space. 3. Hypergeometric - is when there is replacement when choosing from the sample space. 4. Uniform - every outcome has the same probability (i.e. rolling a dice or flipping a coin). 5. Gamma - is when observing a Poisson dist. and you are waiting for the th change to occur. 6. Chi-Squared - when a Gamma dist. with =1/2 or =2. For limits: 1/ = 0; ex Variations of the expected value: E[x1 + x2 + … + axn] = x1+ x1 + … + xn E[ax1 + ax2 + … + axn] = ax1+ ax1 + … + axn E[h(x1) l(x2)] = E[h(x1)]E[l(x2)] All by independence Math 310 1 Probability and Statistics - Final Exam Study Guide Variations of the variation: Var(x1 + x2 + … + xn)= Var(x1) * Var(x2) * … * Var(xn) Var(ax1 + ax2 + … + axn)= a2Var(x1) * a2Var(x2) * … * a2Var(xn) All by independence Moment generating functions: Mx+y(t) = Mx(t)My(t) M(t) of a Uniform dist. is P(e1t + e2t + …+ ent) when n R and p is the probability. To normalize us x-/ 1. X=N(0,1), then X2=2(1) 2. X=N(,), then (x-/)2=2(1) X is a r.v. and defines something it is not a probability. pdf is for continuous distributions so we must integrate. pmf is for discrete distributions. * y=n x= 2y=n2 2x=2/n * Confidence Intervals for Condition What to use X is Normal and 2 is known; n N Z/2 and X is not Normal and 2 is known; n Z/2 and X is any dist. and 2 is unknown; n Z/2 and s X is not Normal and 2 is known; n t/2 and X is any dist and 2 is unknown; n t/2 and If there is a ‘*’ by the table all the x’s in the table are x-bar not just x. When converted a Normal dist. into a chi-squared if you use the sample mean instead of the mean you will loose 1 degree of freedom. To find P(X1>X2) when X1 and X2 are N(_,_) then use P(Y>0) where Y=X1-X2 and is N(1- 2,21+22) CLT: Regardless of what type of distribution you start with, if n is large, the closer W will be to standard normal. Wn=(Xn-)/(/n) So Wn ~ N(0,1) when n is large enough. Type of Distribution Value of n that make it a good approx. Any 25 Normal Any value, it is exact Approx. Normal = 2 or 3 The pdf is cont. and symmetric = 4 or 5 When using CLT, is the dist is not continuous don’t forget to add or subtract .5 (depending on what is to be included in the probability). For T distributions r = n-1. For F distributions use F(r1,r2) is it is 1/F(r1,r2) then use F(r2,r1). Also P(F > 1/3.97) = P(1/F < 3.97) Get the a and b values for CI of 2 by using Table IV but if minimal CI are needed use Table X. Get the a and b values for CI of 2x/2y by using Table VIII [F(m-1,n-1)]. Math 310 2 Probability and Statistics - Final Exam Study Guide For one-sided confidence intervals for proportions the end points are: [-1, 1]. To determine sample size we need to know , the max error of the estimate, it can be found with the formula, if it is not given to us: = Z/2 /n and of course the formula for sample size: n=(( Z/2 )/)2 If we don’t have the value of phat, we will assume phat to be equal to 1/2., so phat(1-phat) will be at its max. Critical region is when to reject H0. H0 is the null hypothesis (no change hypothesis). H1 is the alternate hypothesis (researcher’s hypothesis). Type I error (): rejecting H0 when H0 is true (also the significance level of the test). Type II error (): rejecting H1 when H1 is true. For test of hypotheses for one or two proportion(s) use the value in the back cover that is given, but if p=p0 and p<p0 the inequality gets switched and Z becomes –Z; if p=p0 and pp0 then Z become |Z| and Zbecome Z/2. For test of hypotheses for variances or mean (no matter which one there are) if the inequality is < the inequality will also get switched, and is the inequality is the we or (+) the 2 equations where each equation has a different inequality, and the Z or t will be come Z/2 or t/2. P-value is P(t or Z is of a great degree). Math 310 3 Probability and Statistics - Final Exam Study Guide

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posted: | 1/19/2010 |

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