Content Notes

Question 1

Sample Space Ω=

Answer 1

{a,b,c,…}, events are defined e.g. 1,2,3 or S=success, F=failure

Question 2

Discrete r.v:

Answer 2

A specific value in a set e.g. {1,2,3}

Question 3

Continuous r.v:

Answer 3

A continuous value in a set e.g. (0,1)

Question 4

fx(x)=

Answer 4

P(X=x)

Question 5

Fx(x)=

Answer 5

P(X≤x)

Question 6

P(X≤b)=

Answer 6

P(X≤a) + P(a<X≤b)

Question 7

Fx(b)=

Answer 7

Fx(a) + P(a<X≤b)

Question 8

P(a<X≤b)=

Answer 8

Fx(b) - Fx(a)

Question 9

P(X≤10)=

Answer 9

Fx(10)

Question 10

P(X<10)=

Answer 10

Fx(9)

Question 11

2 conditions of a Bernoulli trial:

Answer 11

A succession of random independent experiments that must have,
1. Only 2 outcomes, success or failure
2. Constant probability, p

Question 12

Bernoulli Trial pmf:

Answer 12

Y={0, if failure (1-p); 1, if success (p)

Question 13

Probability function of a Bernoulli trial:

Answer 13

fY(y)=P(Y=y)={p, if y=1; 1-p, if y=0

Question 14

Binomial Distribution fx(x)=

Answer 14

P(X=x)=(n x)p^x(1-p)^n-x

Question 15

P(A|B)=

Answer 15

P(AnB)/P(B)

Question 16

Bayes Theorem:

Answer 16

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

Question 17

Partition Rule for a Bernoulli trial:

Answer 17

P(S)=P(S|T)P(T) + P(S|Tc)P(Tc)

Question 18

Likelihood rule:

Answer 18

L(p;x)=P(X=x)=(n x)p^x(1-p)^n-x for 0≤p≤1

Question 19

Maximum Likelihood Estimate:

Answer 19

dL(p;x)/dp =0, p=p̂, p̂=x/n

Question 20

Maximum Likelihood Estimator:

Answer 20

To find the maximum value of p we take L(p) and derive it to get dL/dP, the max value occurs when p=0.

Question 21

Var(p̂)=

Answer 21

Var(x/n)=1/n^2 Var(X) = 1//n^2 np(1-p) = p(1-p)/n

Question 22

Var(aX)=

Answer 22

a^2(Var(X))

Question 23

Var(X)=

Answer 23

np(1-p) = E(X^2)-(E(X))^2

Question 24

Var(aX+b)=

Answer 24

a^2(Var(X))

Question 25

E(X)=

Answer 25

np

Question 26

sd(X)=

Answer 26

sqrt(Var(X))=sigma(x)

Question 27

Standard error of p̂=

Answer 27

sqrt((p̂(1-p̂))/n)

Question 28

What is the 95% confidence interval?

Answer 28

The true value is within 2 standard errors 95% of the time.

Question 29

Distribution mean:

Answer 29

E(X)=Sum of xP(X=x) = Sum of xfx(x) = EX = ux

Question 30

Binomial Distribution mean:

Answer 30

E(X) = Sum of x(n x)p^x(1-p)^n-x = np

Question 31

What is a Monte Carlo simulation?

Answer 31

A statistical technique that predicts possible outcomes of an event.

Question 32

What are the 2 basic ideas of the Monte Carlo simulation?

Answer 32

1. Repeat an experiment r times 2. The relative frequency of the value observed is the approximate probability of that value, e.g. If we see X=10 when N=50, then our p=0.2.

Question 33

Finite population of size N, variance=

Answer 33

1/N Sum of (xi-u)^2

Question 34

Population mean=

Answer 34

(1/N) Sum of xi

Question 35

Estimator for Sn^2:

Answer 35

1/n Sum of (xi-xbar)^2

Question 36

Estimator for Sn-1^2

Answer 36

1/n-1 Sum of (xi-xbar)^2

Question 37

geometric distribution

Answer 37

P(X=x)= (1-p)^x*p