Probability

Question 1

When exclusive? When independent?

Answer 1

A U B = A + B
A n B = AB

Question 2

Bayesian

Answer 2

B|A = B (A|B) / A

Question 3

Stars and bars

Answer 3

p+q-1 choose q
q balls with p boxes

Question 4

Binomial
Binomial distribution P(X=r) = ?

Answer 4

sum (n choose r) q^r p^n-r

Question 5

(n choose r)=?+?

Answer 5

(n-1, r) + (n-1, r-1)

Question 6

P(3H in 6 coin tosses)

Answer 6

(6 choose 3) p^3 (1-p)^3

Question 7

P(birthdays)

Answer 7

P365, r / 365^r

Question 8

Discrete probability Variance equation

Answer 8

E(x2) - E(x)2

Question 9

Discrete probability Var(aX ± bY) =
Discrete probability Var(X+a) =

Answer 9

a2 Var(x) + b2 Var(x)
Var(x)

Question 10

Binomial distribution Mean, variance?

Answer 10

np
np(1-p)

Question 11

When is poisson used?

Answer 11

successes is unlimited

Question 12

Poisson P(X=r) = ?

Answer 12

λ^r e^-λ / r!

Question 13

Poisson Mean, variance?

Answer 13

λ
λ

Question 14

Poisson Derive mean, derive variance

Answer 14

sum r f(r) = e^-λ sum(λ^r / (r-1)!) = e^-λ λ sum(λ^s / s!) = λ

Question 15

PDF Give uniform distribution
PDF Mean, variance?

Answer 15

1/b - a if between a and b
a+b/2
(b-a)² / 12

Question 16

Normal distribution f(x) = ?
Mean, variance?

Answer 16

1/ σ√2π e^ - ((x - μ)/σ√2)²
N(μ, σ²)

Question 17

Normal distribution Derive mean, derive variance

Answer 17

E(x) = … 1/√π int μe^-y² dy = μ

Question 18

Normal distribution CDF draw

Answer 18

s

Question 19

Normal distribution CDF formula

Answer 19

1/ √2π int e^ - (z/2)²

Question 20

Normal distribution 𝜎, 2𝜎, 3𝜎

Answer 20

0.683
0.954
0.997

Question 21

State CLT

Answer 21

if xi samples are taken from a distribution, the distribution of mean is well approximated by normal distribution as n is large