Ch1 Probability Fundamentals Flashcards
1
Q
Discrete probability characterized by:
A
pmf = p(x)
2
Q
pmf = p(x) satisfies :
A
Kolmogorov Axioms
- p(x) = Pr(X=x)
- p(x) ge 0 for all x
- sum (p(x)) =1 over all x
3
Q
cdf =
A
cumulative distribution function
F(x) = Pr(X le x)
F(x) = sum(p(X le x))
4
Q
E[X]=
A
sum(x*p(x)) over all x (continuous case use integrals)
5
Q
E[X]= words
A
mean X, long run average, 1st absolute moment
6
Q
Var(X)=
A
= E[(X - E[X] )^2) ]
= E[X^2] - E[X]^2
7
Q
Var(X)= (computational form)
A
E[X^2] - E[X]^2
8
Q

A

9
Q

A

10
Q

A

11
Q

A

12
Q

A

13
Q

A

14
Q
rth absolute moment
A
E[Xr]
15
Q
rth central moment
A
E[(X-µ)r]
16
Q
Var(X)=
A
σ2 = E[(X-µ)2]
17
Q

A

18
Q

A

19
Q
marginal pmf
A

20
Q
R.V.’s are independent iff
A

21
Q

A

22
Q
Cov(X,Y)=
A

23
Q

A

24
Q

A

25


26
Corr(X,Y)=

27
Bernoulli distribution

28
Binomial distribution

29
Poisson distribution

30
Normal distribution

31


32


33
CLT

34
normal approximation to binomial

35
continuity correction for continuous approx to discrete distribution
eg. normal approx to binomial

36
chi-square

37
t-distribution

38


39
Ô is an unbiased estimator of O iff
E[Ô] = O
40
Ô is a weakly consistent estimator of O iff
for any small positive constant, €,
Pr( |Ô - O| \< € ) → 1 as n→inf
also called convergence in probability
Ô→P O
41
IF B(Ô) → 0 and Var(Ô)→0 as n→inf
then Ô is a consitent estimator of O
42
MSE(Ô) =
E[( Ô - O )2] = Var(Ô) + Bias2(Ô)
43
Relative efficiency
R.E. = RE(S1, S2) = MSE(S2) / MSE(S1)
44
RE(X, Y) \< 1
X is less efficient than Y
45
Markov's inequality
Pr( X \> a) le E[X]/a
46
Chebyshev’s inequality (words)
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
47
Pr(|X-A|=\>KY)
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.
48
Slutsky’s theorem
IF Ô1 →P O1
and
Ô2 →P O2
Then sum and product also converge
49
E[S2]=
Var[S2]=
remember (n-1)/σ2 \* S2 = chi-squared (df = n-1)
E[S2] = σ2
Var(S2) = σ4 / (n-1)2