Probability

Question 1

Define a sample space

Answer 1

For an experiment, the sample space is the set of all outcomes

Question 2

Define an event

Answer 2

A subset of the sample space

Question 3

Define equally likely for a finite sample space

Answer 3

P(A) = |A|/|Ω| for all A

Question 4

Define a permutation

Answer 4

The number of ways to order disinguishable objects. For n objects, there are n! different ways.

Question 5

Define the binomial coefficient

Answer 5

The number of orderings of m of one object and n - m of another is the binomial coefficient, which is equal to n!/[(m!)(n-m)!]

Question 6

Define a probability space

Answer 6

A triple (Ω, F, P) where Ω is the sample space, F is a collection of events (subsets of Ω) and P is a function from F to the real numbers

Question 7

Give the axioms of F

Answer 7

F1: Ω ∈ F
F2: If A ∈ F, then A complement ∈ F
F3 if Ai ∈ F for all i ≥ 1, then the union of Ai for i to infinity ∈ F

Question 8

Give the axioms of P

Answer 8

P1: P(A) ≥ 0 for A in F
P2: P(Ω) = 1
P3: If Ai ∈ F for i ≥ 1and all Ai and Aj are disjoint for all i ≠ j, then the probability of the union of the individual events = the sum of the probabilities of the individual events.

Question 9

Define conditional probability

Answer 9

The conditional probability of A given B is: P(A|B) = P(A∩B)/P(B)

Question 10

Define a partition

Answer 10

{B_1, B_2, …} partitions Ω if Ω = U(B_i) and (B_i)n(B_j) = {} whenever i ≠ j.

Question 11

Give the law of total probability

Answer 11

For a partition of Ω: B1, B2, … with PBi) > 0 for each i > 0, P(A) = P(A|Bi)xP(Bi), summed from i to n.

Question 12

Give Bayes’ Theorem

Answer 12

For a partition of Ω: B1, B2, … with PBi) > 0 for each i > 0, then:
P(Bk|A) = P(A|Bk)xP(Bk)/P(A)

Question 13

Define independence of two events

Answer 13

A and B are independent if P(A∩B) = P(A)P(B)

Question 14

Define independence of a family of events

Answer 14

A family {Ai,i∈I} of events is independent if P[n(i∈J)Ai] = Product of Ai for all i∈J for all finite subsets J of I

Question 15

Define a discrete random variable

Answer 15

A discrete random variable X on a probability space is a function X:Ω->R where:
Im(X) {X(ω),ω∈Ω}a is a countable set
For each X in R, {ω∈Ω:X(ω)=x} is in F

Question 16

Define the probability mass function

Answer 16

The function R->[0,1]

Question 17

Define the Bernoulli distribution

Answer 17

X~Ber(p) if P(X=1) = p and P(X=0) = 1 - p

Question 18

Define the Binomial distribution

Answer 18

X~Bin(n,p) if P(X=k) = nCk.p^k.(1-p)^(n-k) for all k from 0 to n.

Question 19

Define the Geometric distribution

Answer 19

X~Geom(p) if P(X=k) = p(1-p)^(k-1) for k>1.

Question 20

Define the Uniform distribution

Answer 20

On a finite set {x1,x2,x3,…,xn} P(X=xi)= 1/n for i = 1, 2, 3, …, n

Question 21

Define the Poisson distribution

Answer 21

X~Poisson(λ) if P(X=k) = (λ^k*e^-λ)/k!

Question 22

Define expectation of a discrete variable

Answer 22

E[X] = x.P(X=x), summed for all x in the indexing set

Question 23

Define independence of discrete random variables

Answer 23

Two discrete random variables X and Y are independent if for all x and y in the reals, the events {X=x} and {Y=y} are independent

Question 24

Define joint distributions

Answer 24

Suppose X and Y are discrete random variables on (Ω,F,P). The joint probability mass function of X and Y is p_(x,y):R^2->[0,1]

Question 25

Give Stirling’s approximation

Answer 25

n! ∼ (2π)^(1/2)xn^(n + 1/2)xe^(−n)

Question 26

Define marginal distribution

Answer 26

For X, Y with joint probability mass function p_XY, the marginal distribution of X is given by the summing all the values of y in p_XY(x,y)

Question 27

Define the conditional probability mass function

Answer 27

The conditional probability mass function of X given B is p_(X|B) (x) = P({X=x}∩B)/P(B)

Question 28

Define conditional expectation

Answer 28

The conditional expectation of X given B, E[X|B] = x(p_x|B)x, summed for all x in ImX

Question 29

Give the law of total probability for expectations

Answer 29

E[X] = E[X|Bi]P(Bi), summing i in I, defined whenever E[X] exists

Question 30

Give the formula for linearity of expectation

Answer 30

E[X+Y] = E[X] + E[Y]

Question 31

Define variance

Answer 31

The variance of a discrete random variable X is Var(X) = E[(X-E[X])^2], provided both of these expectations exist.

Question 32

Define standard deviation

Answer 32

The square root of variance

Question 33

Define covariance

Answer 33

For discrete random variables X and Y, cov(X,Y) = E[XY] - E[X]E[Y]

Question 34

Define the probability generating function

Answer 34

G_x(s) = (p_k)(s)^k, summed for all k from one to infinity.

Question 35

Define a random variable

Answer 35

A function X: Ω -> R such that {X ≤ x} := {ω ∈ Ω : X(ω) ≤ x}

Question 36

Define a cumulative distribution function

Answer 36

The cumulative distribution function of a random variable X is the function F: R -> [0,1] defined by F(x) = P(X ≤ x)

Question 37

Define a continuous random variable X

Answer 37

A random variable whose cdf can bewritten as the integral of fx(u)du from -infinity to x.

Question 38

Define a probability density function of X

Answer 38

The function fx: R -> R in the integrand of a cdf of a continuous random variable

Question 39

Define a continuous uniform distribution

Answer 39

X ~ Unif([a,b]) if X is a random variable with pdf fx(x) = 1/(b-a) for a ≤ x ≤ b

Question 40

Define an exponential distribution

Answer 40

For λ > 0, X ~ Exp(λ) if X is a random variable with pdf fx(x) = λe^(-λx) for x ≥ 0

Question 41

Define a normal distribution

Answer 41

For μ in the reals and σ^2 > 0, X ~ N(μ, σ^2) if X is a random variable with pdf fx(x) = 1/(2πσ^2)*exp(-(x-μ)^2/2σ^2)

Question 42

Define the expectation of a continuous random variable

Answer 42

E[X] = integral of xfx(x)dx for all x from - infinity to infinity, provided that the integral is less than infinity.

Question 43

Define the joint cdf of two continuous distributions

Answer 43

F_XY(x,y) = P(X≤x,Y≤y)

Question 44

Define jointly continuous

Answer 44

X and Y are jointly continuous if F_X,Y(x,y) = the double integral of f_X,Ydydx, with each from -∞ to x and y respectively.

Question 45

Define independence of jointly continuous random variables

Answer 45

Independent if f_X,Y(x,y) = f_X(x)f_Y(y)

Question 46

Define a random sample

Answer 46

An independent selection of random variables from a distribution

Question 47

Define the sample mean

Answer 47

The sum of the variables in the sample, divided by the number of variables in the sample

Question 48

Give the weak law of large numbers

Answer 48

Probability of |(1/n).(sum of X from 1 to n) - μ| > ε converges to 0

Question 49

Give Markov’s inequality

Answer 49

P(X ≥ t) ≤ E[X]/t

Question 50

Give Chebyshev’s inequality

Answer 50

P(|Z - E[Z]| ≥ t) ≤ var(Z)/t^2