W9/10 - Prob

Question 1

Mutually exclusive

formula and meaning

Answer 1

Pr⁡(A ∪B)=Pr⁡(A)+Pr⁡(B)

no interception

Question 2

set difference formula

Pr⁡(B\A)=??

Answer 2

Pr⁡(B\A)=Pr⁡(B)−Pr⁡(A ∩ B)

Question 3

addition rule for 2 sets then 3 sets

Answer 3

Pr(A∩B) = Pr(A)+Pr(B)−Pr(A∪B)
3 sets is
Pr⁡(A ∪B∪C)=Pr⁡(A)+Pr⁡(B)+Pr⁡(A∩B)−Pr⁡(A∩C)−Pr⁡(B∩C)+Pr⁡(A∩B∩C)

Question 4

Independent events formula

Answer 4

Pr⁡(A∩B)=Pr⁡(A)∗Pr⁡(B)

Question 5

Conditional probability formula

Formula and Bayes’ theorem

Answer 5

Pr⁡(A│B)=Pr⁡(A∩B)/Pr⁡(B)

Pr⁡(A|B)=[Pr⁡(B│A) x Pr⁡(A)]/Pr⁡(B)

Question 6

formula and properties

Expected value (AKA expectation or mean)

What if X and Y are independent?

Answer 6

E(X)=∑k∗Pr⁡(X=k)

E(αX)= αE(X)
E(X+Y)= E(X)+ E(Y)
E(XY) = E(X)E(Y) (independence only)

Question 7

Median

The value m where these two conditions are true

Answer 7

Pr⁡(X≤m)≥1/2 Pr⁡(X≥m)≥1/2

Question 8

Mode

Answer 8

The value g for which Pr(X=g) is greatest.

Question 9

Variance and sd

what if X and Y are independent?

Answer 9

Var(X)=E(X^2 )−[E(X)]^2 =E((X−u)^2 )

Sd(X)=√Var(X)

Var(X+Y)=Var(X)+Var(Y)

Question 10

Chebyshev’s inequality

Answer 10

Pr⁡(|X−E(X)|≥t∗sd(X)) ≤ 1/t^2

the probability that X is at least t standard deviations away from the mean is at most 1/t^2

Question 11

Uniform distribution

Formula and notation

Answer 11

Pr⁡(X=x)=1/(b−a+1)

(can be calulated from mean and var formula)
E(X)=(a+b)/2=median
Var(X)=((b−a+1)^2−1)/12

Notation: X~Unif(a,b)

The uniform distribution gives the same probability to all [a,b], and zero probability to all other integers.

Question 12

What is E(X) and Var(X)

Binomial distribution

Formula and notation

Answer 12

Pr⁡(X=k)=(nCk)*(p^k)* [(1−p)^(n−k)]

E(X)=np
Var(X)=np(1-p)

X~Bin(n,p)

Question 13

What is E(X) and Var(X)

Poisson distribution

Formula and notation

Answer 13

Pr⁡(X=k)=(e^μ ∗ μ^k)/k!

E(X)=μ
Var(X)=μ

X~Poisson(μ)

For rare events, if you are given just mean, it is most likely this.

Question 14

What is E(X) and Var(X)

Geometric distribution

Formula and notation

Answer 14

Pr⁡(X=k)=(1−p)^(k−1) ∗p

E(X)=1/p
Var(X)=(1−p)/p^2

X~Geom(p)

Counting the number of trials needed to get the first success (same condition as Bin)

Question 15

Applications of distributions - coupon collectors problem

Answer 15

Collecting all n different type of coupons.

Let Z=X_1+X_2+…+X_n

P_k=(n−k+1)/n
X_k~Geom((n−k+1)/n)

E(X_k)=1/P_k
E(Z)=E(X1+X2+...+X3)=n(1/1+1/2+...+1/n) (do not need to remember this, can find out if needed)

to get the k-th coupon