Prob A Flashcards
What is a sample space?
The set whose elements each correspond to a possible outcome of the experiment, so that every possible outcome is included
What is an event?
A property that the outcome of the experiment may or may not have. We can identity the elements of the sample space with the relevant property
Give the notation for
i) A and B
ii) A or B
iii) not A
iv) A is a subset of B
What is a probability measure
What does it mean for A and B to be mutually disjoint?
The intersection of A and B is empty
Prove that P(A) + P(Ac) = 1
Prove the relationship between P(A) and P(B) if A is a subset of B
Prove that P(AuB) = P(A) + P(B) - P(AnB)
State the Inclusion-Exclusion formula for events not necessarily disjoint
What is the total number of permutations of {1,2,….,n}
n!
Lemma; Let S be a set having n elements and let 0<=k<=n. The number of subsets of S having exactly k elements is nCk = n!/(k! x (n-k!))
State the Fundamental Multiplication rule
Suppose a procedure has k steps with n1 ways of performing the 1st step, n2 ways of performing the second step, up to nk ways of performing the kth step. Then the total number of different ways is n1 x n2 x …. x nk
Suppose a population of size N=K1 + K2 contains K1 individuals of type 1 and K2 individuals of type 2. If sample of size n is drawn with replacement from this population it contains k1 individuals of type 1 and k2 individuals of type 2. Whats the probability
Suppose a population of size N=K1 + K2 contains K1 individuals of type 1 and K2 individuals of type 2. If sample of size n is drawn without replacement from this population it contains k1 individuals of type 1 and k2 individuals of type 2. Whats the probability
Give the conditional probability of P(B | A) and state the important conditions
State and Prove the multiplication rule for conditional probabilities
When are events a partition of a subspace
State the law of total probability
Prove the total law of probability
State Bayes Formula
Prove Bayes formula
When are events E1,E2,…..,En mutually independent
When for all choices of 1 to n
P(E1 n E2 n …. n En) = P(E1) x P(E2) x … x P(En)
When does a random variable X have the binomial distribution with parameters n and p
Prove the binomial probability distribution
State the law of large numbers for the Binomial distribution
State the binomial to poisson distribution theorem
Prove the conversion of binomial to poission
What is the most likely outcome of a experiment with n repeats and probability of success p
np
State the Theorem for the gaussian approximation to the normal distribution
