Random Variables

Question 1

Discrete Random Variable

Answer 1

X with values x = 1,2,…

EX = sum (x_i)( P(X = x_i) )

Var(X) = E(X^2) - (EX)^2

Question 2

Uniform Random Variable

Answer 2

EX = (x_1 + … + x_n) / n

Var(X) = (x_1^2 + … + x_n^2) / n - (EX)^2

Question 3

Bernoulli Random Variable

Answer 3

aka indicator random variable represented by an event A with probability p

I_A = 1 if A occurs or 0 otherwise

E(I_A) = p
Var(I_A) = p - p^2 = p ( 1 - p )

Question 4

Binomial Random Variable

Answer 4

Binomial(n,p)

– number of successes in n independent trials

pmf = P(X = i) = nCi p^i (1-p)^n-i

EX = np

Var(X) = np - np^2 = np(1-p)

Question 5

Poisson Random Variable

Answer 5

Poisson(lambda)

pmf = P(X = i) = lam^i / i! e^ - lam

EX = lambda

Var(X) = lambda

Question 6

Poisson Approximation of binomial

Answer 6

when n is large and p is small and

lambda = np is of moderate size

P(X = i) –> lam^i / i! e^(-lambda)

as n–> infinity

Question 7

Geometric Random Variable

Answer 7

Geometric(p) counts the number of independent trials before the first success

pmf = P(X = n) = p(1-p)^n-1

EX = 1/p

Var(X) = 1-p / p^2

P(X > n) = (1-p)^n

P(X > n + k given X > k) = P(X > n)

Question 8

Continuous Random Variables…

Answer 8

no longer discuss pmf like we did with discrete random vars

now use Distribution Functions given by
F = F_X = P(X < x) = \int_(-infinity,x] f(s) ds

so, F’(x) = f(x)

where f(s) is the density function of X and F is the distribution function

int _ (-inf,inf) f(x) dx = 1

EX = int (-inf,inf) x f(x) dx

Var(X) = E(X^2) - (EX) ^2

Question 9

Continuous Uniform Random Variable

Answer 9

density:
f(x) = 1 / b-a if x \in [a,b]
0 otherwise

EX = a+b / 2

Var(X) = (b-a)^2 / 12

Question 10

Continuous Exponential Random Variable

Answer 10

Exponential(lambda)

used for waiting time for an event to occur

density: f(x) = lambda e^(- lambda x) for x >= 0
0 otherwise

EX = 1 / lambda

Var(X) = 1 / lambda^2

P(X >= x) = e^(- lambda x)

Memoryless property:
P(X >= x + y given X >= y) = e^(- lambda x)

Question 11

Normal Random Variable

Answer 11

Normal(mu, sig^2)

density: f(x) = ( 1 / sig\sqrt(2pi) ) e^( (x-mu)^2 / 2sig^2 )

EX = mu

Var(X) = sig^2

Question 12

normal approximation for binomial

Answer 12

binomial is approximately normal when p is fixed and n is large

then,

P( S_n - np / sqrt(np(1-p)) < x) –> PHI(x)

as n–> infinity

Question 13

Joint pmf for discrete random variables

Answer 13

X,Y are discrete random vars

joint pmf = P(X=x,Y=y)

marginal probability:

P(X = x) = sum_(all y) P(X=x,Y=y)

P(Y = y) = sum_(all x) P(X=x,Y=y)

Question 14

Independence of random variables

Answer 14

discrete case:
independent if the joint pmf is the product of the marginal pmfs

P(X = x, Y = y) = P(X = x) P(Y = y)

continuous case:
independent if the joint density is the product of the marginal densities

f(x,y) = f_X(x) f_Y(y)

Question 15

joint density functions for continuous random variables

Answer 15

two continuous vars X, Y

let f(x,y) be the joint density function

the marginal density functions are given below:

f_X(x) = int_(-inf,inf) f(x,y) dy

f_Y(y) = int_(-inf,inf) f(x,y) dx

Question 16

Conditional pmf for discrete random variables

Answer 16

P(X = x given Y=y) = P(X=x,Y=y) / P(Y=y)

Question 17

Conditional density for conditional random variables

Answer 17

f_X(x given Y=y) = f(x,y) / f_Y(y)

where f(x,y) is joint density and f_Y(y) is the marginal density