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| Poisson Approximation formula |
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Definition
| (e^-u*u^k)/k! where k is the number of successes |
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| When should the Poisson Approximation be used? |
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Definition
| When n is large and p is very small |
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| Sampling with replacement formula |
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Definition
(n choose g)*[G^gB^b/N^n]
You can also think of this technique in terms of a Bernoulli trial: (n k)p^kq^n-k. You should get the same answer |
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| Sampling without replacement formula |
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Definition
((G g)(B b))/(N n)
where G is the good elements of the population, g is the good elements of the sample, B is the bad elements of the population, b is the bad elements of the sample, N is all the elements of the population and n is all the elements of the sample
Note: Bernoulli trials does not apply here because you're not replacing objects |
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Term
| Random Variable definition |
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Definition
| A random variable is a real valued function defined on a probability space. |
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Term
| Expectation of a random variable X, E(X) definition |
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Definition
| the sum (x is an element of the range of X) of xP(X = x) |
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Term
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Definition
| The range of a random variable is the set of all the values it takes |
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Term
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Definition
represents the probability that the random variable X equals a certain number x on a subset of the outcome space. So to figure it out, think of all the possible subsets of outcomes you can have and then count the number of subsets that matches what you're looking for.
For example, P(X=5) where X(a,b) = a+b and a and b represent the numbers you get from rolling two die, the probability is 4/36. There's 4 subsets where a+b = 5: (1, 4), (4, 1), (2, 3), (3,2)
There's also 36 total subsets so it's 4/36 |
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| The probability of any subset B of the range is gotten by adding up the probabilities of all the singleton subsets that it contains |
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Definition
| For example: P({2,3,4} for the random variable example (P{2,3,4}) = 1/36+2/36+3/36 = 6/36 Remember the range is all the possible values X can take so you're basically finding the probability that X can be either a 2, or 3, or 4 |
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Term
| Taking functions of random variables |
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Definition
You can define an additional random variable by taking functions of X.
For example, g(s) = s^2 and is defined on the RANGE of X. Then Z = g(X) = X^2
The range of Z would then be all the numbers in the range of X squared |
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Term
| The probabilities of a random variable X and an additional random variable defined on the range of X |
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Definition
They're the same!
Continuing the X((a, b)) = a+b example...
P(X=6) = 5 because there's 5 ways you can roll a die and get 6 as the answer.
If Z = g(X) where g = s^2, then the probability that P(Z=36) also equals 5 because there's 5 ways you can still get a sum that gets you 36. Essentially there's 5 things (these things being outcomes of X) that get you 36 |
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Term
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Definition
For two different random variables, joint distribution is defined as P(X=x and Y=y) or P(X=x, Y=y).
It is defined on the space of all possible pairs (x, y) where x is in the range of X and y is in the range of Y.
P(X=x, Y=y) has to be greater than 0 to be in this space. Means (x, y) has to a possible event.
This is a bit confusing but an element (c, d) is where X (based on its definition - don't forget this!) gives you c and d (based on its definition) gives you d. |
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Term
| Making a joint distribution table for (X, Y) |
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Definition
| Put all the possible values of Y on the side and all the possible values of X on the top. Then (using a coordinate like system) calculate the probability that you'll get an outcome that give you those coordinates based on the definitions of the random variable |
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Term
| Partition Equation for a joint distribution |
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Definition
| Says P(X = a certain value x) = P(X=x and Y=y1) + P(X=x and Y = y2) + P(X=x and y = y3) etc. Essentially you have to add up all the probabilities that give you X=x regardless of what Y is |
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| What does saying X=Y mean? |
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Definition
| It means that X and Y are defined on the same outcome space and that they take the same values on all points of that outcome space. |
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Term
| Independence of random variables |
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Definition
Two random variables X , Y are independent if for any values x,y we have
P(X=x, Y=y) = P(X=x)P(Y=y)
This implies P(Y=y|X=x) = P(Y=y) and
P(XeB,YeC) = P(XeB)P(YeC)
Additionally, E(XY) = E(X)E(Y) |
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Term
| Definition of expectation |
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Definition
E(X) = the sum where x is an element in the range of X (R) of xP(X=x). It can be interpreted as a long-term average
This is analogous to the expected value in Bernoulli trials (np) |
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Term
Let X be the sum of the number on two dice when they are thrown. The range of X is the set {2,3,4,5,6,7,8,9,10,11,12}
What is E(X)? |
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Definition
| 2*1/36 + 3*2/36 + 4*3/36.....etc = 7 |
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Term
| Addition Rule of Expectation |
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Definition
| E(X + Y) = E(X) + E(Y) for any random variables X and Y no matter if they're independent or not |
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Term
| Indicator Functions (Indicator RV) |
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Definition
An indicator function is a random variable defined on an outcome space that only takes the values 0 and 1
So basically you apply the function to a subset of the outcome space and you'll always get either a 0 or 1 for the answer
Denoted by IsubB
Think of it as denoting true or false if the w is in the subset B. Isub(w) = 1 if "some condition about w". The answer will be 1 if w is in B, 0 or not |
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| Indicator function square rule |
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Definition
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Term
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Definition
IsubAIsubB
Represents the indicator function for their intersection |
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Definition
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Definition
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Definition
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Term
| If IsubA is an indicator function, then E(IsubA) = |
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Definition
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Term
| Tail Sum Formula for Expectation |
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Definition
For X with possible values {0,1,...,n},
E(X) = the sum from j = 1 to n of P(X >= j)
Use this when calculating E(X) is difficult |
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Term
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Definition
If X>= 0, then for any number a>0, P(X>=a) <= E(X)/a
In problems, a is the number we're looking for. So if the question asks the probability of getting 80 heads, a = 80 |
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Term
| For random variables X1, X2,...Xn, E(X1, X2,...Xn) = |
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Definition
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Term
| Expectation of a function of X |
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Definition
If g is a function on the range of X, then:
E(g(X)) = the sum of all x on g(x)P(X=x) |
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Term
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Definition
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Term
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Definition
c
This happens when the constant random variable only has the value of c |
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Term
Just for clarification:
X =
E(X) =
which also equals: |
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Definition
Isub(A1) + Isub(A2) + ... + I(subAn)
E(sub(A1))+ E(Isub(A2)) + ... + E(I(subAn)
P(A1) + P(A2) + ... + P(An) |
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Term
| Standard deviation of a random variable X |
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Definition
| the square root of the variance of X |
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Term
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Definition
| = E((X-u)^2) = E(X^2) - E(X))^2 |
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Term
| True or false: E(X) is the mean (u) |
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Definition
| true. Remember u = np and we already established that E(X) is a long term average |
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Term
| Chebychev's Inequality (simple form) |
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Definition
Let X be any random variable. Let u denote E(X) and let sigma be the standard deviation of X:
then for any number k>0, the probability that the value of X wil be found k or more standard deviations away from its expected value is <= 1/k^2 |
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Definition
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Term
| Var(X1 + X2 + ... + Xn) if X1 + X2 + ... + Xn are independent random variables = |
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Definition
| Var(X1) + Var(X2) +...+ Var(Xn) |
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Term
| Two random variables have the SAME DISTRIBUTION if |
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Definition
they take the same values with the same probabilities.
This implies that the random variables will also have the same expectation and variance |
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Term
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Definition
Let S(subn) be the sum and A(subn) be the average of n independent random variables with the same distribution as X.
Then:
E(Sn) = nE(X) = np
E(An) = E(X) = p
SD(Sn)= (square root of n)*SD(X) = square root of npq
SD(An) = SD(X)/square root of n = square rot of pq/square root of n |
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Term
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Definition
the sum from 0 to infinity of lambda^k = 1/1-lambda
P(of some number n) = q^n-1*p |
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Term
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Definition
| P(A wins) = P(A)/P(A)+P(B) |
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Term
| Mean and standard deviation for a geometric distribution |
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Definition
E(X) = 1/p
STD = square root of (q/p^2) |
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