What does A compliment look like using a venn diagram?
What does A U B, and A B and look like using a venn diagram?
Translate the following set notation into event laungage.
A ∩ B.
A and B occur
What is statistical independence?
RVs are said to be independent if the outcome of any one Xi does not influence and is not influenced by and other RV Xj.
What is identical distribution?
RVs are identically distributed if their density functions are identical in the sense that f(X1| theta) = f(X2|theta) = f(Xn|theta)
What are the 3 conditions for a random experiment
1. All possible distinct outcomes are known a-priori
2. In any particular trial the outcome is not known a-priori but there exists a discernible regularity of occurrence associated with these outcomes.
3. The experiment can be repeated under identical circumstances.
Define Sample Space
All possible distinct outcomes of an experiment.
For flipping a coin S={H,T}
For SAT scores S={200,210,220,...,780,790,800}
For reaction time to a stimulus S would consist of all positive numbers i.e. S=(0, OO)
Define Event
An event is a collection of possible outcomes of an experiment, that is any subset of S (including S itself).
Let A be an event, a subset of S. We say the event A occurs if the outcome of the experiment is in A.
Subset
A is a subset of S ( A C S or A C S) if every element of A is also in S.
Union
Union of sets A and B ( A U B) is all the elements that are in both or either sets.
Intersection
Intersection of A and B ( A ∩ B) is all the elements are are in BOTH A and B.
Define Compliment
The compliment of set A (A^{c})=A' is all the elements that are not a part of A. If A{0,1,2,3,} and S{0,1,2,3,4,5} A^{c} would be {4,5}
What are some alternate notations for compliment.
A^{c}=A'=B-C, OR B^{c}=B'=A-B
Empty Set
Empty set (Ø) is the impossible event. It does not occur. It contains no elements.
Define Event Space
An event space F is a set whose elements are the events of interest as well as the related events i.e. those obtained by combining the events of interest using set theoretic operations. When ∩, U, or the compliment, is applied to any elements of F, the result is also an element of F.
What is the definition of a field?
A non-empty class A of sets is called a field of sets iff it is closed under:
1. Finite unions
2. Finite intersections
3. Compliments
Define the 3 conditions for a set to be a field
A collection F of subsets S is said to be a field if it satisfies 3 conditions:
1. S ε F
2. If A ε F then A^{c} ε F.
3. If A, B ε F then ( A U B) ε F.
Mathematically describe what two mutually exclusive (or disjoint) events A and B look like
A∩B=Ø, or the intersection of A and B has no elements.
For a continuous random variable, the probability that it takes a specific value is ?
Zero, probability for such a variable is measurable only over a given range or interval, such as (a,b).
A random variable may be either ?
Discrete or continuous
Define a discrete RV
A discrete RV takes on only finite (or countable infinite) number of values. An example would be the sum of numbers on 2 dice thrown. X can be 2,3,4,5,6,7,8,9,10,11, or 12. I.e. if its range is a countable subset of the R.
Define a continuous RV
A continuous RV is one that can take on any value in some interval. The height of an individual is a continuous RV in the range of say 60-65 inches depending on the precision in the measurement. I.e. if its range of values is any uncountable subset of the R.
The probability density function of a continuous variable is defined as the ...
Derivative of the (cumulative) distribution function.
What is a random variable?
A function that is a mapping from the original sample space S to the real numbers. Or a function that attaches numbers to all elements of S in a way that preserves the event structure of F.
If the exp is tossing 2 dice, X could be the sum of the numbers.
If the exp is tossing a coin 25 times, X could be the number of heads in 25 tosses.
The nature of the RV depends crucially on the size of the field in question. If F is small, being a RV w.rt. F is very restrictive.
What are the 3 things that make up a probability set space?
(S, F, P( )). The sample space, the field, and the probability set function.
What 2 things are needed for a random trial?
1. The set up for the experiment remains the same for all trials.
2. The outcome in one trial does not effect that of another. (Independence)
What is Bayes' Formula?
P(A|B)= P(B|A) * P(A) / P(B) for P(B)>0
What is a density function?
It describes the probability for each x ϵ S. The density function has the values on the X-axis and their associated probabilities on the Y-axis. By definition the probabilities sum to 1.
What are the 4 properties of the (cumulative) distribution function?
1. Fx(.) is non-decreasing
2. Fx(.) is right continuous
3. Lim as x -> OO : = Fx(OO) = 1
4. Lim as x -> -OO : = Fx(-OO) = 0
The integral of the PDF is the ?
(Cumulative) distribution function
The derivative of the (C) DF is the ?
Probability density function
Axiom of Finite additivity: If A and B are disjoint (mutually exclusive), then P( A U B) =
P(A) + P (B)
What is the sample space of : tossing a coin 2 times?
HH, HT, TH, TT
What is the sample space of the lifetime in hours of a particular brand of light bulb?
S=(0, OO)
P(A|B)= (probability of A conditional on B or prob A given B
P(A B) / P(B)
Probability of S (sure event) is
1
Probability of the null set is
0
The probability of A happening (or not happening) must be ?
0<=A=<1
Prob of A^{c} [A compliment] is?
P(A^{c})= 1-P(A)
If A C B (a is in b) then the P(A) must be ?
<= P(B)
P(AUB) = ?
4 answers
1. P(A) + P(A^{c} B)
2. P(B) + P(AB^{c} )
3. P(A^{c} B) + P(AB^{c}) + P(AB)
4. P(A) + P(B) - P(AB)
Define disjointedness (mutually exclusive):
Two sets are disjoint iff they have no elements in common (or AB=0, intersection of a and b is zero. A U B is a union of disjointed sets. If you're looking at a venn diagram, the circles don't overlap.
Define set equality :
Two sets are equal iff A C B and B C A. In which case A=B
What is the product rule for conditional probability ?
What is CP2 ? In mutual exclusive events, if the occurrence of event A implies the occurrence of one of the Bi then what is the prob A?
P(A) =sum from I to J of P(A|Bi)P(Bi)
Expand using DeMorgan's Law:
A (B U C)= ?
A U (B C)= ?
A (B U C)= (A ∩ B) U (A ∩ C)
A U (B C)= (A U B) ∩ (A U C)
Expand using DeMorgan's Law:
(A U B)^{c}= ?
(A ∩ B)^{c}= ?
(A U B)^{c}= A^{c }∩ B^{c}
(A ∩ B)^{c}= A^{c} U B^{c}
What is bayes' rule?
P(B|A)=?
P(A|B)*P(B) / P(A)
Translate the set notation into event laungage.
A - B
A occurs and B does not occur
Translate the set notation into event laungage.
A U B - A ∩ B
A or B, but not both, occur.
Translate the set notation into event laungage.
A - (B U C)
A occurs, and B and C do not occur.
Translate the set notation into event laungage.
A C B
If A occurs, then B occurs but if B occurs then A need not occur. A is a subset of B.
Translate the set notation into event laungage.
A ∩ B= 0
If A occurs, then B does not occur or if B occurs then A does not occur.
If A ∩ B=0, then A and B are disjoint sets, or mutually exclusive.
True or false U and ∩ are commutative laws. Meaning the order does not matter. A U B= B U A
True: U and ∩ are commutative laws.
What are De Morgan's Laws
(A U B)^{c }= ?
(A ∩ B)^{c} =?
(A U B)c = A^{c} ∩ B^{c}
(A ∩ B)c = A^{c} U B^{c}
Let U be the set of people solicited for a contribution to a charity. All the people in U were given a chance to watch a video and to read a booklet. Let V be the set of people who watched the video, B the set of people who read the booklet, C the set of people who made a contribution.
(a) Describe with set notation: "The set of people who did not see the video or read the booklet but who still made a contribution"
(b) Rewrite (a) using demorgan's law.
A. (V U B)^{c} ∩ C.
B. V^{c} ∩ B^{c} ∩ C.
If A ∩ B =0, then n(A U B)= ? [n= number of units, instead of saying a probability]
(Inclusion-Exclusion Principle part 2)
n(A) + n(B).
If A C B, then n(A) ? n(B). [n= number of units, instead of saying a probability]
(Inclusion-Exclusion Principle part 3)
n(A) =< n(B).
If n(A U B) =
(Inclusion-Exclusion Principle part 1)
n (A) + n (B) - n(A ∩ B).
A total of 35 programmers interviewed for a job; 25 knew FORTRAN, 28 knew PASCAL, and 2 knew neither languages. How many knew both languages?
(Inclusion-Exclusion Principle)
Let F be the group of programmers that knew FORTRAN, P those who knew PASCAL. Then F ∩ P is the group of programmers who knew both languages. By the Inclusion-Exclusion Principle we have n(F U P) = n(F)+n(P) - n(F∩ P): That is, 33 = 25+28 - n(F ∩ P): Solving for n(F ∩ P) we find n(F ∩ P) = 20.
n(A * B) = n(A) * n(B):
Cardinality of the Cartesian product of two finite sets.
n(A) * n(B):
What is 0!
1
Calculate the following:
10!/7!
10!/7!=10*9*8=720
What is the formula for P(n,k) permutations?
How many five-digit zip codes can be made where all digits are different?
= 30,240
What is the formula for combinations of n objects taken k at a time?
What is the alternative notation for C(n,k)?
From a group of 5 women and 7 men, how may different committees consisting of 2 women and 3 men can be formed?
C (5,2) C(7,3) = 350 different possible committees.
The chess club has 6 members. In how many ways
(a) can all 6 line up for a picture?
(b) can they choose a president and a secretary?
(c) can they choose 3 members to attend a regional tournament with no regard to order?
(a) P (6,6) = 6! = 720 different ways
(b) P (6,2) = 6! / (6-2)! = 6!/4! = 6*5= 30 ways
(c) C (6,3) = 6! / 3!(6-3)! = 720/36 = 20 different ways
Given n objects of which n1 indistinguishable objects of type 1, n2 indistinguishable objects of type 2, ..., nk indistinguishable objects of type k where n1 + n2 + ... + nk = n: Then the number of distinguishable permutations of n objects is given by: ?
How many different rearrangements of the letters DECIEVED are there?
8 letters, d twice, e three times= 8! / 2!3!= 3360
How many ways can you arragne 2 red, 3 green, and 5 blue balls in a row?
(2+3+5)! / 2!3!5!= 2520 ways
P (A|B) = ?
P(A|B)= P (A ∩ B) / P(B)
Probability of A given B= number of outcomes corresponding to A & B divided by the number of outcomes of B.
Bayes' Formula is useful in finding the "reverse" of a given conditional probabilty. I.e. if you know A|B what is B|A?
State Bayes' Formula for P(B|A).
P(B|A)= P (A∩B)/ P(A) = P (A|B)P(B) / P(A|B)P(B) + P(A|B^{c})P(B^{c}).
Using two events A and B; define A as the union of two events related to B.
*Bayes' Formula
P(A)= P(A∩B) + P(A∩B^{c})
= P(A|B)P(B) + P(A|B^{c})P(B^{c}) .
State the general version of Bayes' Formula for events H_{i} and A. P(H_{i}|A)?
P(H_{i}|A)= =
True or false: When using conditional probabilities, you can use relative frequencies like "smokers died at twice the rate as non-smokers" and you don't need exact numbers.
True : The results will be the same as long as the relative frequencies are the same. Smokers died at .4 and non at .2 is the same as smokers died at .8 and non at .4.
True or false: If two events are independent then P(A|B)=P(A)
True. Two events are independent if P(A|B)=P(A) or there is no overlap in the 2 circle using a venn diagram. If there is any chance that both events could happen, then they are not independent.
True or False: If A and B are independent then so are A and B^{c}.
True
A combination is a ________ where you don't care about the __________.
Permutation, order
True or false: the order matters in a permutation.
True
True or false: order matters in a combination
False
How do you change a permutation into a combination?
Divide the permutation formula n!/(n-k)! by the number of item pairings or k. So in order to divide the formula you multiply the denominator by k and you get n!/k(n-k)!
What is the prob of getting exactly 3 out of 8 heads on a coin flip?
It's the combinations that would give you three divided by the total number of outcomes. Or C(8,3) / 2^8. So 8!/8(8-3)! Divided by 2^8.
A random variable is not a variable in the traditional sense because you never ______ for it. It's more like a __________. It maps a random process into a __________. It transforms random processes into a PDF .
Solve, function, number
What is the binomial distribution? And what does it look like ?
The binomial distribution is the number of successes in a sequence of n independent yes/no experiments. It looks like the discrete case of the normal distribution.
Describe what a 3 cases of a binomial distribution would look like. Where one has a low prob of success, the second and equal prob of success, and the third a high prob of success.
What probability technique do you use to create a binomial distribution?
Permutations as the numerator, and the total number of cases as the denominator.
True or false: a binomial distribution will always be symmetric like a normal distribution.
False, it will only be symmetric if the probabilities of success and failure are equal. If failure is more likely, it will be skewed left and if success is more likely, it will be skewed right.
What does the Poisson distribution model?
It is used to model the number of events occurring within a given time interval.
What are 3 applications of the Poisson distribution?
1. Number of phone calls coming into a call center during a given period.
2. Number of deaths per year from a given age group.
3. Number of insurance claims during a given time period.
What does the poisson distribution look like?
What is the formula for the poisson distribution.
What is the general form for P(A1A2....An | E)?
P(A1|E) * P(A2|A1E) * P(An|A1...An-1E)
What is the following expression equal to: P(A U B U C)?
P(A U B U C)= P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC)
Using conditional probabilities and the multiplication rule: what is the P(ABC)?
P(ABC)= P(A) * P(B|A) * P(C|AB)
How are a Bernoulli distribution and a binomial distribution related?
A Bernoulli distribution is just like a binomial distribution but it consists of only 1 trial.
Describe the geometric distribution.
The geometric distribution is used in the case where you do an activity until you have a success. An example would be a person flips a coin until he gets a heads, or a basketball player shoots until he makes a basket.
What is the formula for the pdf of the geometric distribution?
P(X=r) = (1-p)^{r-1} p, for r=1, 2, 3, ....
What does the geometic distribution look like?
Describe the negative binomial (or Pascal) distribution.
It is the distribution where the experiment consists of independent Bernoulli trials, with a constant probability of success, and we are interested in finding how long it will take for the rth success to occur.
What does the negative binomial (or Pascal) distribution look like?
The negative binomial distribution looks like a normal, but with a slighly longer right tail, and a sharper sloped left tail.
If you have a negative binomial distribution where you only need one success, what do you have.
A negative binomial dist. with only one success becomes the geometric distribution.
What is the pdf formula for the exponential distribution?
What does the pdf of an exponetial distribution look like?
What would be some of the uses for a exponential distribution?
1. How long do we need to wait until a customer enters a shop?
2. How long will a piece of machinery work before breaking down?
3. How much time will pass before an earthquake occurs in a given region?
How are the exponential and poisson distributions related?
Exponential is until 1 event happens. When the event can occur more than once and the time elapsed between two successive occurrences is exponentially distributed and independent of previous occurrences, the number of occurrences of the event within a given unit of time has a Poisson distribution.
What is a uniform (rectangular) distribution?
The uniform distribution is a type of probability distribution in which all outcomes are equally likely.
What does the PDF of the uniform distribution look like?
What is the formula for the pdf of the uniform distribution?
What is the formula for the cdf of the uniform distribution?
What is this the cumulative distribution function (cdf) of ?
Uniform distribution
What does the pdf of the normal distribution look like?
What does the cdf of the normal distribution look like?
What is the formula for the pdf of the normal distribution?
What is the moment generating function for the normal distribution?
Why do we use moment generating functions?
Because if you use MGFs then you only have to take derivatives and not integrals of those functions to find the moments in the functions of interest. Like mean, variance, skewness, and kurtosis.
What is the formula for E[x]?
E[x]=x*f(x)
What is the formula for var(x)?
Var(x)=E(x^{2})-E(x)^{2}
What is the relationship between the variance and standard deviation of x
Standard deviation is the square root of the variance.
What is the general formula for the moment generating function Mx(t)?
Mx(t)=E(e^{xt})
What the var(X+Y)=?
if x and y are not independent.
Var (X+Y)=Var(X)+(Var(Y)+2Cov(X,Y)
Var(X-Y)=?
If x and y are not independent?
Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)
Var(X+Y)=?
Var(X-Y)=?
If X and Y are independent.
Var(X+Y)=Var(X-Y)=Var(X)+Var(Y)
Two events A, B have no intersections. In other words, A∩B =∅. Does this mean that A, B are independent?
A∩B =∅ doesn't mean that A, B are independent.A∩B =∅ merely means that A, B are mutually exclusive. Mutually exclusive events may be affected by a common factor, in which case A, B are dependent.
Two random variables X ,Y have zero correlation (i.e. ρ X ,Y = 0 ). Are X ,Y independent?
If X ,Y are independent, then , 0 X Y ρ = . However, the reverse may not be true; zero correlation doesn’t automatically mean independence. X ,Y may have a non-linear relationship (such as Y = X 2 ). As a result, even, ρ_{x,y}=0 , X ,Y may not be independent from each other.
If P( A∩B∩C) = P( A) P(B) P(C), does this mean that A, B,C are independent?
P(A∩B∩C) = P( A) P(B) P(C)does not mean that A, B,C are independent.
A, B,C are independent if the following conditions are met:
P(A∩B) = P( A) P(B)
P(A∩C) = P( A) P(C)
P(B∩C) = P(B) P(C)
P(A∩B∩C) = P( A) P(B) P(C)
How do you find the percentile of a random variable X?
For a random variable X , p -th percentile (donated as xp ), means that Pr(X<=xp )=p% <-> F(xp)=p% <-> Pr(X>xp)=1-p%
How do you find the median of a random variable x?
Median is the 50th percentile. X_{50}
If F(x)=, then F(X_{50})=.5=
How do you calculate the mode of a distribution?
The Mode is the most observed. f(x) is maxed at the mode when d/dx f(x) [at the lower integration limit]=0.
So if f(x)=1/9(4-2x), f(x) is zero at x=2. So the mode is 2.