Topic 4 - Discrete Random Variables

Question 1

What is P(X = x) formally known as

Answer 1

• Probability distribution function or probability mass function

Question 2

What does the probability distribution/mass function show

Answer 2

• The probabilities for all the possible outcomes
• Can be shown algebraically, graphically or in a table

Question 3

What is E(x) for a discrete random variable

Answer 3

• sum of x * P(x)

Question 4

What is the cumulative distribution function denoted as

Answer 4

• F(x0)
• P(X <= x0)

Question 5

What is the variance for a discrete random variable

Answer 5

• E(X - mu)^2 = sum of (x - mu)^2 * P(x)

Question 6

What is the standard deviation of a discrete random variable

Answer 6

• sqrt of sum of (x - mu)^2 * P(x)

Question 7

What are E(a) and Var(a) equivalent to

Answer 7

• E(a) = a
• Var(a) = 0

Question 8

What are E(bX) and Var(bX) equivalent to

Answer 8

• E(bX) = b * E(X)
• Var(bX) = b^2 * Var(X)

Question 9

If Y = a + bX what are E(Y) and Var(Y) equivalent to

Answer 9

• E(a + bX) = b * E(X) + a
• Var(a + bX) = b^2 * Var(X)

Question 10

What is a bernoulli distribution

Answer 10

• A bernoulli random variable X can be thought of as an indicator of success taking two values
• X = 1, succes, X = 0 if failure

Question 11

When is a random variable said to have a bernoulli distribution

Answer 11

• if P(X = 1) = p and P(X = 0) = 1-p

Question 12

What is E(X) and Var(X) for a bernoulli distribution

Answer 12

• E(X) = p
• Var(X) = p(1 - p)

Question 13

What is the binomial distribution formula

Answer 13

• P(X = k) = n! / k!(n-k)! * p^k * (1-p)^n-k

Question 14

How is the binomial distribution denoted

Answer 14

• Bin(n,p)

Question 15

What does the binomial distribution look like to be considered a bernoulli distribution

Answer 15

• Bin(1,p)

Question 16

What is E(X) and Var(X) of a binomial distribution

Answer 16

• E(X) = np
• Var(X) = np(1-p)

Question 17

When would a poisson distribution be used

Answer 17

• Where we are counting the number of success in a particular space or interval of time

Question 18

What are the 4 assumptions of a poisson distribution

Answer 18

• An event can occur any number of times in a given time period
• The probability of an event occuring is proportional to the length of the time period
• The rate of occurance is constant
• Events occur independently

Question 19

What is the poisson distribution formula

Answer 19

• P(X = k) = e^-λ * λ^k / k!

Question 20

What is lamda in a poisson distribution

Answer 20

• A constant that specifies the the average of occurances for a particular time/space

Question 21

What is E(X) and Var(X) for a poisson distribution

Answer 21

• E(X) = λ
• Var(X) = λ

Question 22

How is the poisson distribution denoted

Answer 22

• Po(λ)

Question 23

What is the general rule for a poisson rule when changing time intervals

Answer 23

• if X ~ Po(λ) on 1 unit interval then Y ~ Po(kλ) on k unit intervals

Question 24

When can the poisson distriubtion be used to approximate binomial probabilities

Answer 24

• When the number of trials n is large
• When the probability p is small
• When np is moderate (preferably np <= 7)

Question 25

Show mathmatically how the poisson approximates the binomial

Answer 25

- Bin(n,p) -> λ = np - Po(np) - P(X = k) = e^ -np * (np)^k / k!

Question 26

What is a joint probability mass function (PMF)

Answer 26

- Used to express the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y

Question 27

How can a joint probability mass function be represented mathmatically

Answer 27

- P(x,y) = P(X = x n Y = y) - where n is the intersection

Question 28

How are the marginal probability mass functions, P(X) and P(Y), calculated using the joint function P(x,y)

Answer 28

- P(X) = sum of P(x,y) for all y - P(Y) = sum of P(x,y) for all x

Question 29

How do you calculate the conditional probability mass function using marginal and joint functions

Answer 29

- P(y | x) = P(x,y) / P(x) - P(x | y) = P(x,y) / P(y)

Question 30

When are jointly distributed random variables, x and y, said to be independent

Answer 30

- P(x,y) = P(x) * P(y)

Question 31

How is covariance calculated for joint variables x and y (discrete random variables)

Answer 31

- covariance is the expected value of (X - mu of x) * (Y - mu of y) - Cov(X,Y) = E[(X - mu of x)(Y - mu of y)] = sum of all of sum of all y of (x-mu of x)(y-mu of y) * P(x,y)

Question 32

What does covariance measure

Answer 32

- Measures the strength of the linear relationship between two variables - If the two variables are statistically independent the covariance between them is 0

Question 33

How is the correlation between X and Y calculated

Answer 33

ρ = Corr(X,Y) = Cov(X,Y) / S.D of x * S.D of y

Question 34

What do different values of ρ indicate

Answer 34

- ρ = 0 no linear relationship between X and Y - ρ > 0 positive linear relationship between X and Y - ρ < 0 negative linear relationship between X and Y

Question 35

If W = aX + bY, what are E(W) and Var(W) for portfolio analysis

Answer 35

- E(W) = E(aX + bY) = a * mu of x + b * mu of y - Var(W) = Var(aX + bX) = a^2 * Var(X) + b^2 * Var(Y) + 2 * a * b * Cov(X,Y) - Var(W) = Var(aX + bX) = a^2 * Var(X) + b^2 * Var(Y) + 2 * a * b * S.D of x * S.D of y * Corr(X,Y)