# Math 310 â€“ 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

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