Discrete random variables Flashcards
Define a discrete random variable
A random variable that can only take certain values in a finite or countable set
Define the probability mass function of a DRV
p(x) = P(X=x)
State two properties of pmfs
1) all p(x) ≥ 0
2) Sum of all p(x) = 1
Define the kth moment of X
E(Xk)
Similarly, the kth moment of X about a is E((X-a)k) (where a is a real number)
Give the expectation of E(XY) if X and Y are independent
E(XY) = E(X)E(Y)
Note: this property does not prove independence
Define the case for when rvs X and Y are independent
if P(X=x,Y=y) = P(X=x)P(Y=y) for all x and y, X and Y are independent
Give and describe Chebyshev’s inequality
Give and describe the weak law of large numbers
This law demonstrates that the long-run average of a random variable is very unlikely to be far from its expectation, as the variance of the long-run average gets smaller and smaller.
State the expectation of the Bernoulli distribution
p, where p is the probability of success
State the variance of the Bernoulli distribution
p(1-p), where p is the probability of success
Describe the Bernoulli random variable X and give the proper notation
X is defined as the number of successes we have if we perform the experiment.
Describe the Binomial random variable X and give its proper notation
X is defined as the number of successes we have counted after n independent Bernoulli trials have taken place
Give the probability mass function of the Binomial distribution
Note that the cdf for the binomial distribution is the sum of the individual probabilities.
State the E(X) for the binomial distribution
np
State the Var(X) of the binomial distribution
np(1-p)
Describe the geometric random variable X and give its proper notation
X is defined as the number of independent Bernoulli trials with the same probability of success up to and including the first success. X* may be used to denote “the number of failures until the first success”, and X* = X - 1
Give the pmf for the geometric distribution
State the E(X) for the geometric distribution
State the Var(X) of the geometric distribution
Describe the Poisson random variable X and give the proper notation
X is defined as the number of events we had recorded on one such surface we had available
Give the pmf for the poisson distribution
State the E(X) of the poisson distribution
State the Var(X) of the Poisson distribution
Describe how the Poisson distribution can be used to approximate the binomial distribution
- This applies if n is large, p is small and np is moderate
- μ = np
Describe the negative binomial random variable X and give the proper notation
X is defined as the number of independent Bernoulli trials with the same probability of success, until we have r successes.
This is equivalent to r.v. X* for the number of faliures until the rth success, where X* = X - r
Give the pmf of the negative binomial distribution
State the E(X) of the negative binomial distribution
State the Var(X) of the negative binomial distribution
State the E(X) of the hypergeometric distribution
State the Var(X) of the hypergeometric distribution
This does not need to be memorised
Describe the hypergeometric random variable X and give its proper notation
X is defined as the number of units in the sample (of size n and that we selected without replacement) that have the characteristic
Give the pmf of the hypergeometric distribution
Where we have N units, from which M share the same characteristic. We select n units from the population of N without replacement
Describe how the hypergeometric and binomial distributions can be approximated.
When the ration of n/N is small, the hypergeometric (n,M,N) distribution may be approximated by the binomial (n,p) where p = M/N
