Topic 4 - Discrete Random Variables Flashcards
What is P(X = x) formally known as
- Probability distribution function or probability mass function
What does the probability distribution/mass function show
- The probabilities for all the possible outcomes
- Can be shown algebraically, graphically or in a table
What is E(x) for a discrete random variable
- sum of x * P(x)
What is the cumulative distribution function denoted as
- F(x0)
- P(X <= x0)
What is the variance for a discrete random variable
- E(X - mu)^2 = sum of (x - mu)^2 * P(x)
What is the standard deviation of a discrete random variable
- sqrt of sum of (x - mu)^2 * P(x)
What are E(a) and Var(a) equivalent to
- E(a) = a
- Var(a) = 0
What are E(bX) and Var(bX) equivalent to
- E(bX) = b * E(X)
- Var(bX) = b^2 * Var(X)
If Y = a + bX what are E(Y) and Var(Y) equivalent to
- E(a + bX) = b * E(X) + a
- Var(a + bX) = b^2 * Var(X)
What is a bernoulli distribution
- A bernoulli random variable X can be thought of as an indicator of success taking two values
- X = 1, succes, X = 0 if failure
When is a random variable said to have a bernoulli distribution
- if P(X = 1) = p and P(X = 0) = 1-p
What is E(X) and Var(X) for a bernoulli distribution
- E(X) = p
- Var(X) = p(1 - p)
What is the binomial distribution formula
- P(X = k) = n! / k!(n-k)! * p^k * (1-p)^n-k
How is the binomial distribution denoted
- Bin(n,p)
What does the binomial distribution look like to be considered a bernoulli distribution
- Bin(1,p)
What is E(X) and Var(X) of a binomial distribution
- E(X) = np
- Var(X) = np(1-p)
When would a poisson distribution be used
- Where we are counting the number of success in a particular space or interval of time
What are the 4 assumptions of a poisson distribution
- 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
What is the poisson distribution formula
- P(X = k) = e^-λ * λ^k / k!
What is lamda in a poisson distribution
- A constant that specifies the the average of occurances for a particular time/space
What is E(X) and Var(X) for a poisson distribution
- E(X) = λ
- Var(X) = λ
How is the poisson distribution denoted
- Po(λ)
What is the general rule for a poisson rule when changing time intervals
- if X ~ Po(λ) on 1 unit interval then Y ~ Po(kλ) on k unit intervals
When can the poisson distriubtion be used to approximate binomial probabilities
- When the number of trials n is large
- When the probability p is small
- When np is moderate (preferably np <= 7)
Show mathmatically how the poisson approximates the binomial
- Bin(n,p) -> λ = np
- Po(np)
- P(X = k) = e^ -np * (np)^k / k!
What is a joint probability mass function (PMF)
- 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
How can a joint probability mass function be represented mathmatically
- P(x,y) = P(X = x n Y = y)
- where n is the intersection
How are the marginal probability mass functions, P(X) and P(Y), calculated using the joint function P(x,y)
- P(X) = sum of P(x,y) for all y
- P(Y) = sum of P(x,y) for all x
How do you calculate the conditional probability mass function using marginal and joint functions
- P(y | x) = P(x,y) / P(x)
- P(x | y) = P(x,y) / P(y)
When are jointly distributed random variables, x and y, said to be independent
- P(x,y) = P(x) * P(y)
How is covariance calculated for joint variables x and y (discrete random variables)
- 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)
What does covariance measure
- Measures the strength of the linear relationship between two variables
- If the two variables are statistically independent the covariance between them is 0
How is the correlation between X and Y calculated
ρ = Corr(X,Y) = Cov(X,Y) / S.D of x * S.D of y
What do different values of ρ indicate
- 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
If W = aX + bY, what are E(W) and Var(W) for portfolio analysis
- 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)
