Module 9 Vocab

Question 1

Random Variable (textbook def)

9.1

Answer 1

a random variable on (Ω, Pr) is a function
X: Ω →ℝ

Question 2

Random Variable (Alex def)

9.1

Answer 2

a function that maps each outcome in the sample space to a real number

Question 3

The set of values taken by X

9.1

Answer 3

Val(X) = {x ∈ ℝ| ∃w ∈ Ω X(w) = x}

Question 4

Another way to say “the set of values taken by X”

9.1

Answer 4

returned

Question 5

X = x

9.1

Answer 5

the set of all outcomes that are mapped to the real number “x”

Question 6

Pr[X = x]

9.1

Answer 6

the probability of the outcomes that are mapped to “x”
probability of event X = x

Question 7

Distribution of r.v. X

9.1

Answer 7

f: Val(X) → [0,1] where f(x) = Pr[X = x]
Ex in green die space: f(1) = Pr[X=1] = 1/6

Question 8

Probabilities add up to 1 (symbols)

9.1

Answer 8

Σ x∈Val(X) Pr[X = x] = 1

Ex: flip coin space Pr[X=H] + Pr[X=T] = 1/2 + 1/2

Question 9

Probabilities add up to 1 (words)

9.1

Answer 9

“if you sum all the probabilities of the events that the random variable X=x, for all the x in the valus taken by the r.v. X, the answer is 1”

Question 10

Events (X=x) for x∈Val(X) are…

9.1

Answer 10

pairwise disjoint

Question 11

Ux∈Val(X) (X = x) =

9.1

Answer 11

Ω

Question 12

Σx∈Val(X) Pr[X = x] =

9.1

Answer 12

Pr[Ω] = 1

Question 13

Σ x∈Val(X) Pr[X = x] = 1

9.1

Answer 13

“if you sum all the probabilities of the events that the random variable X=x, for all the x in the valus taken by the r.v. X, the answer is 1”

Question 14

Uniform Distribution

9.1

Answer 14

f: {v_1,…,v_n} → [0,1] f(v_i) = 1/n i = 1,…,n

Question 15

Given (Ω, Pr), a r.v. U: Ω → ℝ is uniform with these values (v_1,…,v_n) when:

9.1

Answer 15

Val(U) = {v_1,…,v_n}
and Pr[U = v_i] = 1/n i = 1,…,n

Question 16

Val(U) = {v_1,…,v_n}
and Pr[U = v_i] = 1/n i = 1,…,n

9.1

Answer 16

uniform r.v.

Question 17

f: {v_1,…,v_n} → [0,1] f(v_i) = 1/n i = 1,…,n

9.1

Answer 17

uniform distribution

Question 18

Bernoulli r.v. with parameter p

9.1

Answer 18

Given (Ω, Pr), an r.v. X: Ω → ℝ
with Val(X) = {0, 1}
and Pr[X = 1] = p

Question 19

Val(X) = {0, 1}
and Pr[X = 1] = p

9.1

Answer 19

Bernoulli r.v. with parameter p

Question 20

Distribution of Bernoulli r.v.

9.1

Answer 20

f: {0,1} → [0, 1]
f(1) = p f(0) = 1 - p

Question 21

f: {0,1} → [0, 1] f(1) = p f(0) = 1 - p

9.1

Answer 21

Bernoulli distribution with parameter p

Question 22

How many ways can 3 dice sum up to 5?

9.1

Answer 22

Stars and bars (5-3) + 3 - 1 choose 3 - 1 = 6

Question 23

Stars and bars with restriction of each die

9.1

Answer 23

From “n” subtract how many are used to meet condition.
Ex. if there are 5 die then n - 5
Ex. if there are 10 die, then n - 10

Question 24

Expectation/Expected Value

9.2

Answer 24

average (mean) value returned by a r.v.

Question 25

E[X]

9.2

Answer 25

the expectation of a r.v. X

Question 26

Expected Value of X (symbols)

9.2

Answer 26

E[X]

Question 27

E[X] way 1

9.2

Answer 27

x∈Val(X) Σ x ⋅ Pr[X = x]

value returned times its weight

Question 28

E[X] way 2

9.2

Answer 28

w∈Ω Σ X(w) ⋅ Pr[w]

value mapped to by outcome times probability of the outcome

Question 29

E[D]

9.2

Answer 29

3.5

remember: D is the number shown by a fair dice

Question 30

Expectation of a uniform r.v.

9.2

Answer 30

that takes the values v1,…,vn:
v1 +…+ vn / n

Question 31

Expectation of the Bernoulli r.v.

9.2

Answer 31

recall Val(X) = {0, 1} and Pr[X=1] = p and Pr[X=0] = 1 - p
E[X] = 1 ⋅ Pr[X=1] + 0 ⋅ Pr[X=0]
= 1⋅ p + 0 ⋅ (1 - p)
= p

Question 32

p

9.2

Answer 32

Expectation of the Bernoulli r.v.

Question 33

E[C] = c

9.2

Answer 33

expectation of a constant r.v. proposition

Question 33

Expectation of a constant r.v.

9.2

Answer 33

consider the r.v. C: Ω → ℝ such that for all outcomes w∈Ω we have C(w) = c
E[C] = c

Question 34

Proof of equivalence of the two defs of expectation biconditional statement

9.2

Answer 34

w ∈ [X = w] iff X(w) = x

Question 35

What is X=x?

Answer 35

an event

Question 36

How to use uniform r.v.

Answer 36

state “n” and “v_i = i for i = #,…,#”
remember Pr[X =v_i] = 1/n for i=1,…,n

Question 37

How to use Bernoulli r.v.

Answer 37

state 2 outcomes and give p

Question 38

Anagrams formula

Answer 38

a x b where a is number of b’s
(a1 +…+ an)! / a1! x … x an!

Question 39

Sum of two r.v.’s

Answer 39

(X+Y)(w) = X(w) + Y(w)
add the value that X maps the outcome to and the value that Y maps the outcome to

Question 40

Example of sum of two r.v.’s

Answer 40

S(7) = G(2) + P(5)

Question 41

Scalar multiplication of a r.v.

Answer 41

(cX)(w) = c X(w)

Question 42

Tricky thing to remember about LOE +

Answer 42

for something like A - B you can do A + (-1)B

Question 43

Linearity of Expectation

Answer 43

only for r.v.’s on the same space
version 1: E[cX1 +…+ cXn] = cE[X1] +…+ cE[Xn]

version 2: E[X1 +…+ Xn] = E[X1] +…+ E[Xn]

Question 44

LOE example rolling fair die independently r times

Answer 44

E[W] = E[D1] +…+ E[Dr]
because rolls are independent we can assert that the probability distribution of each roll is the same as the probability distribution of a single die roll which we know E[D] = E[Di] = 3.5 for i = 1,…,r
Thus E[W] = 3.5r

Question 45

What do you need before you use an indicator r.v.?

Answer 45

an event!

Question 46

Indicator r.v.
(3 things)

Answer 46

let Ia be the indicator r.v. for event a
Ia(w) = 1 if w is in a and 0 if w is not in a
IT’S BERNOULLI

Question 47

E[indicator r.v. of event A] =

Answer 47

Pr[IA = 1] = Pr[A]

Question 48

An outcome with k H’s has probability

Answer 48

p^k times q^n-k
where q = 1 - p

Question 49

The expectation of the indicator r.v. is equal to

Answer 49

the probability of its event

Question 50

In order to be able to apply Pr[Hi] to all H_i’s what do we need to know?

Answer 50

the flips/tosses/blahs are I-N-D-E-P-E-N-D-E-N-T

Question 51

On average means we need which concept?

Answer 51

Expectation!

Question 52

Remember if events are independent what can we do to their probabilities?

Answer 52

multiply them!

Question 53

Pr[success followed by failure] =

Answer 53

p(1-p)