Module 10 Vocab

Question 1

R.v.’s X⊥Y if:

10.1

Answer 1

defined on the same Ω
∀x ∈ Val(X)
∀y ∈ Val(Y)
(X=x)⊥(Y=y)

Question 2

Example of independent r.v.’s in green-purple space

10.1

Answer 2

G returns number shown on green die, P returns purple
For any g,p ∈ [1..6]:
Pr[(G=g)∩(P=p)] = (1/6)(1/6) = 1/36 = Pr[G=g] ⋅ Pr[P=p]

Question 3

Pairwise and Mutual Independence

10.1

Answer 3

analogous to events

Question 4

Independence of Indicators and Events (words)

10.1

Answer 4

“for events A,B in same space, their indicator r.v.’s are independent iff the events themselves are independent”

Question 5

Independence of Indicators and Events (symbols)

10.1

Answer 5

Let A,B be two events in (Ω,Pr)
I_A ⊥ I_B iff A⊥B

Question 6

How do you prove iff statements?

10.1

Answer 6

Prove one way
Then prove the other way

Question 7

Independence of indicators and events (extension)

10.1

Answer 7

I_A ⊥ I_B iff A⊥B
can be extended to mutual independence of an arbitrary number of r.v.’s

Question 8

By def, I_A ⊥ I_B tells us

10.1

Answer 8

(I_A = 1) ⊥ (I_B = 1)
(I_A = any #) ⊥ (I_B = any #)

Question 9

By def of indicator r.v. we know

10.1

Answer 9

(I_A = 1) = A and (I_B = 1) = B

Question 10

Expectation of indicator r.v.
E[I_X] =

10.1

Answer 10

Pr[I_X = 1] = Pr[X]

Question 11

How many cases does one indicator r.v. have?

10.1

Answer 11

0 and 1
or x and ˉx

Question 12

IDD Bernoulli trials “performed independently” assume…

10.1

Answer 12

that the Bernoulli r.v. are mutually independent

Question 13

Constant r.v.’s are ________ of any r.v.

10.1

Answer 13

independent

Question 14

Independence of Constant r.v.’s

10.1

Answer 14

for any (Ω, Pr) and any r.v. X: Ω→ℝ and any c∈ℝ, the constant r.v. C: Ω→ℝ defined by ∀w∈Ω C(w) = c is independent of X

Question 15

What do we know about Val(C) of constant r.v. C:Ω→ℝ

10.1

Answer 15

Val(C) = {c} so we also know (C = c) = Ω

Question 16

μ

10.2

Answer 16

mean = expectation = E[X]

Question 17

Deviation of X from its mean

10.2

Answer 17

X - μ, also a r.v.

Question 18

Reminder about a r.v. whose values are constant

10.2

Answer 18

has that same constant as its expectation
E[μ] = μ

Question 19

Variance of a r.v.

10.2

Answer 19

Var[X] = E[(X-μ)²]
where μ = E[X]

Question 20

Standard Deviation of a r.v.

10.2

Answer 20

σ[X] = √Var[X]

Question 21

μ

10.2

Answer 21

Mean

Question 22

σ

10.2

Answer 22

Standard Deviation

Question 23

σ²

10.2

Answer 23

Variance

Question 24

Variance (alternative formula)

10.2

Answer 24

Let X be a r.v. defined on (Ω, Pr)
Var[X] = E[X²] - μ²
where ∀w∈Ω, X²(w) = ((X(w))²)

Question 25

E[X + 2μ] | 10.2

Answer 25

By LOE and expectation of constants = E[X] + 2μ

Question 26

if X is an r.v. that returns positive values, then the ----- of X² is the values ----- with the ---- probabilities | 10.2

Answer 26

distribution squared same

Question 27

If Val(D)=[1..6], then Val(D²) = | 10.2

Answer 27

Val(D²) = {1, 4, 9, 16, 25, 36}

Question 28

Distribution of # shown by fair die | 10.2

Answer 28

X = {1, 2, 3, 4, 5, 6} each with probability 1/6

Question 29

Bernoulli r.v. Expectation | 10.3

Answer 29

E[X] = p

Question 30

X²(w) = 1 iff | 10.3

Answer 30

X(w) = 1

Question 31

X²(w) = 0 iff | 10.3

Answer 31

X(w) = 0

Question 32

If X is Bernoulli... | 10.3

Answer 32

so is X²

Question 33

If X is Bernoulli, then E[X] = | 10.3

Answer 33

E[X²] = p

Question 34

Variance of Bernoulli r.v. with parameter Pr[X=1] = p | 10.3

Answer 34

Var[X] = p(1-p)

Question 35

Bernoulli Expectation and Var | 10.3

Answer 35

different expectation, same variance | Ex: p = 1/3 and q = 2/3

Question 36

Var[cX] = | 10.3

Answer 36

c² Var[X]

Question 37

c² Var[X] = | 10.3

Answer 37

Var[cX]

Question 38

if X⊥Y, then Var[X + Y] | 10.3

Answer 38

Var[X+Y] = Var[X] + Var[Y]

Question 39

Var[X+Y] = Var[X] + Var[Y] iff | 10.3

Answer 39

E[XY] = E[X] ⋅ E[Y]

Question 40

X⊥Y ⇒ | 10.3

Answer 40

E[XY] = E[X] ⋅ E[Y]

Question 41

Variance for sum of 2 dice in green-purple space | 10.3

Answer 41

S = G + P and G⊥P, so Var[S] = Var[G] + Var[P]

Question 42

Binomial r.v. (definition) | 10.4

Answer 42

parameters: n∈ℕ and p∈[0,1] Val(B) = [0..n] and ∀k ∈ [0..n] Pr[B=k] = (n choose k) p^k (1-p)^n-k

Question 43

Binomial r.v. with parameters: n∈ℕ and p∈[0,1] | 10.4

Answer 43

Val(B) = [0..n] and ∀k ∈ [0..n] Pr[B=k] = (n choose k) p^k (1-p)^n-k

Question 44

Distribution of Binomial r.v. B | 10.4

Answer 44

also called binomial with parameters n and p

Question 45

Example of binomial r.v. | 10.4

Answer 45

probability of event "k successes observed" during n IDD Bernoulli trials with probabilty of success p and B r.v. that returns number of successes observed

Question 46

What are expectation and variance completely determined by? | 10.4

Answer 46

the distribution

Question 47

--- and --- are completely determined by the distribution | 10.4

Answer 47

expectation and variance

Question 48

Don't forget: indicator r.v.'s are | 10.4

Answer 48

BERNOULLI!

Question 49

reminder: expectation of indicator r.v. E[I_A] = | 10.4

Answer 49

Pr[A] = p

Question 50

Expectation of Binomial r.v. | 10.4

Answer 50

E[B] = np

Question 51

Pairwise Independence and Variance | 10.4

Answer 51

If r.v.'s X1,...,Xn are pairwise independent, then Var[X1+...+Xn] = Var[X1] +...+ Var[Xn]

Question 52

If r.v.'s X1,...,Xn are pairwise independent, then | 10.4

Answer 52

Var[X1+...+Xn] = Var[X1] +...+ Var[Xn]

Question 53

If the events are mutually independent, then their indicator r.v.'s... | 10.4

Answer 53

are also mutually independent

Question 54

Reminder: variance of Bernoulli r.v.'s with parameter p = | 10.4

Answer 54

p(1-p)

Question 55

Variance for Binomial r.v. | 10.4

Answer 55

Var[B] = np(1-p)

Question 56

X,Y are two r.v.'s on (Ω, Pr), their product, XY, is the r.v. | Self-Paced

Answer 56

(XY)(w) = X(w) ⋅ Y(w) ∀w∈Ω

Question 57

Product of two r.v.'s | Self-Paced

Answer 57

(XY)(w) = X(w) ⋅ Y(w) ∀w∈Ω

Question 58

Product of two r.v.'s (words) | Self-Paced

Answer 58

For every outcome ω in the sample space, you multiply the value that X maps ω to by the value that Y maps ω to in order to obtain the value that the product XY maps ω to

Question 59

X_H * X_T(HH) = | Self-Paced

Answer 59

X_H * X_T(TT) = 0

Question 60

X_H * X_T(HT) = | Self-Paced

Answer 60

X_H * X_T(TH) = 1

Question 61

X_H * X_T is a... | Self-Paced

Answer 61

Bernoulli r.v. with parameter 1/2

Question 62

Correlated r.v.'s | Self-Paced

Answer 62

X,Y r.v. on (Ω,Pr) E[XY] ≠ E[X] ⋅ E[Y]

Question 63

Uncorrelated r.v.'s | Self-Paced

Answer 63

X,Y r.v. on (Ω,Pr) E[XY] = E[X] ⋅ E[Y]

Question 64

Variance distributes over the sum of --- r.v.'s | Self-Paced

Answer 64

uncorrelated

Question 65

Negative Correlation | Self-Paced

Answer 65

Higher A means lower B

Question 66

Covariance | Self-Paced

Answer 66

Cov(X,Y) = E[XY] - E[X]E[Y]

Question 67

Cov(X,Y) = | Self-Paced

Answer 67

E[XY] - E[X]E[Y]

Question 68

two r.v.'s are --- iff their covariance is --- | Self-Paced

Answer 68

uncorrelated, 0

Question 69

for two r.v.'s: all --- r.v.'s are --- but not all --- r.v.'s are ---- | Self-Paced

Answer 69

independent, uncorrelated uncorrelated, independent

Question 70

Correlation is a measure of --- --- | Self-Paced

Answer 70

linear dependence

Question 71

If A⊥B Pr[A|B] = Pr[B|A] = | Self-Paced

Answer 71

Pr[A|B] = A Pr[B|A] = B

Question 72

Alt Var[X+Y] = | Self-Paced

Answer 72

Var[X+Y] = Var[X] + Var[Y] - 2(E[XY] - E[X]E[Y])

Question 73

If X1,...,Xn are pairwise uncorrelated, then: Var[X1+...+Xn] = | Self-Paced

Answer 73

Var[X1] +...+ Var[Xn]

Question 74

Expectation of a product of two r.v.'s E[XY] = | Self-Paced

Answer 74

x∈Val(X)Σ y∈Val(Y)Σ Pr[(X=x)∩(Y=y)] ⋅ x ⋅ y

Question 75

Intersection and Independence, If X⊥Y

Answer 75

Pr[(X=x)∩(Y=y)] = Pr[X=x] ⋅ Pr[Y=y]

Question 76

What 3 things do you need to do in order to use binomial r.v.?

Answer 76

1. make it clear what the n Bernoulli trials are 2. explain why they are IID 3. specify p