Math 310 – Probability and Statistics

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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(EF)\P(F)\
          P(AB)=P(A|B)P(B)
          P(BA)=P(B|A)P(A)
          Events A and B are Mutually Independent if and only if P(AB)=P(A)P(B), otherwise A and
           B are Dependent.
          P(AB)=P(A)+P(B)-P(AB)
          Bayes’ Formula: P(W)=P(B1W)+P(B2W)+P(B3W)
          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(B1W)/P(W)
          Bernoulli f(x)=px(1-p)1-x where xRange, =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] = ax1+ ax1 + … + axn
        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=n2            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 pp0 then Z
    become |Z| and Zbecome 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