Ch1 Probability Fundamentals

Question 1

Discrete probability characterized by:

Answer 1

pmf = p(x)

Question 2

pmf = p(x) satisfies :

Answer 2

Kolmogorov Axioms

1. p(x) = Pr(X=x)
2. p(x) ge 0 for all x
3. sum (p(x)) =1 over all x

Question 3

cdf =

Answer 3

cumulative distribution function

F(x) = Pr(X le x)

F(x) = sum(p(X le x))

Question 4

E[X]=

Answer 4

sum(x*p(x)) over all x (continuous case use integrals)

Question 5

E[X]= words

Answer 5

mean X, long run average, 1st absolute moment

Question 6

Var(X)=

Answer 6

= E[(X - E[X] )^2) ]

= E[X^2] - E[X]^2

Question 7

Var(X)= (computational form)

Answer 7

E[X^2] - E[X]^2

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Answer 12

Question 13

Answer 13

Question 14

r^th absolute moment

Answer 14

E[X^r]

Question 15

r^th central moment

Answer 15

E[(X-µ)^r]

Question 16

Var(X)=

Answer 16

σ² = E[(X-µ)²]

Question 17

Answer 17

Question 18

Answer 18

Question 19

marginal pmf

Answer 19

Question 20

R.V.’s are independent iff

Answer 20

Question 21

Answer 21

Question 22

Cov(X,Y)=

Answer 22

Question 23

Answer 23

Question 24

Answer 24

Question 25

Answer 25

Question 26

Corr(X,Y)=

Answer 26

Question 27

Bernoulli distribution

Answer 27

Question 28

Binomial distribution

Answer 28

Question 29

Poisson distribution

Answer 29

Question 30

Normal distribution

Answer 30

Question 31

Answer 31

Question 32

Answer 32

Question 33

CLT

Answer 33

Question 34

normal approximation to binomial

Answer 34

Question 35

continuity correction for continuous approx to discrete distribution

eg. normal approx to binomial

Answer 35

Question 36

chi-square

Answer 36

Question 37

t-distribution

Answer 37

Question 38

Answer 38

Question 39

Ô is an unbiased estimator of O iff

Answer 39

E[Ô] = O

Question 40

Ô is a weakly consistent estimator of O iff

Answer 40

for any small positive constant, €,

Pr( |Ô - O| < € ) → 1 as n→inf

also called convergence in probability

Ô→^P O

Question 41

IF B(Ô) → 0 and Var(Ô)→0 as n→inf

Answer 41

then Ô is a consitent estimator of O

Question 42

MSE(Ô) =

Answer 42

E[( Ô - O )²] = Var(Ô) + Bias²(Ô)

Question 43

Relative efficiency

Answer 43

R.E. = RE(S₁, S₂) = MSE(S₂) / MSE(S₁)

Question 44

RE(X, Y) < 1

Answer 44

X is less efficient than Y

Question 45

Markov’s inequality

Answer 45

Pr( X > a) le E[X]/a

Question 46

Chebyshev’s inequality (words)

Answer 46

is the theorem most often used in stats. It states that no more than 1/k2 of a distribution’s values are more than “k” standard deviations away from the mean

Question 47

Pr(|X-A|=>KY)

Answer 47

Pr(|X-A|=>KY)<=1/K2,

The absolute value of the difference of X minus A is greater than or equal to the K times Y has the probability of less than or equal to one divided by K squared.

Question 48

Slutsky’s theorem

Answer 48

IF Ô₁ →^P O₁

and

Ô₂ →^P O₂

Then sum and product also converge

Question 49

E[S²]=

Var[S²]=

Answer 49

remember (n-1)/σ² * S² = chi-squared (df = n-1)

E[S²] = σ²

Var(S²) = σ⁴ / (n-1)²