SEM 2 Flashcards
Can the hcf be negative
Yes
What is the prime number formula (Works up to n = 41)?
P(n) = n^2 - n + 41
What are the steps of mathematical induction?
Prove P(N) is true
Prove P(k+1) is true given P(k) is true
Strong induction?
Like mathematical except
P(k + 1) is true when P(N), P(N + 1) … is true
What is fermats little theorem?
a^p = a mod p
a^p-1 = 1 mod p (if p is not factor of a)
Key generation: n = ?
pq
Key Gen: e = ?
e is between 1 and (p-1)(q-1) and
e and (p-1)(q-1) are coprime
Key gen: d = ?
d > 0 such that
de == 1 mod (p-1)(q-1)
What is the public key?
(n,e)
What is private key?
d
How to encrypt?
For each value M
c = M^e mod n
How to decrypt?
M = c^d mod n
How to find inverse of 2x2 matrix?
1/ad - bc (d, -b, -c, a)
How to find inverse of any matrix (not 2x2)
- Write identity matrix next to it
- Turn left side into identity matrix and right side is inverse
What is minor?
Determinant (without working out result)
What is cofactor?
Multiply minor by -1
When does a system of linear equations have a unique solution?
When determinant != 0
When is a matrix invertible?
When the determinant != 0
If a matrix is diagonal what are its eigenvalues?
The diagonal elements
What makes a matrix similar?
B = P^-1 A P
Do similar matrices have the same characteristic polynomials and eigenvalues?
Yes
What makes A diagonalisable?
If there is a diagonal matrix which is similar
Cartesian product?
Set of ordered pairs
Permutation? P(n,k)
n! / (n-k)!
- ordered collection
Combinations? C(n,k)
n! / k! (n-k)!
Probability space consists of?
Sample space and probability function
Conditional probability? A given B
P(A n B) / P(B)
Bayes theorem? (A given B)
P(B | A)P(A) / P(B)
Law of total probability?
P(A) = P(A|B1)P(B1) + … + P(A|Bn)P(Bn)
Expectation value?
Sum of:
x * p(x)
Var?
E(X^2) - E(X)^2
sd?
Root varience
Bernoulli distribution?
- Range of possible values 0 -1
- P(x = 1) = p
- P(X = 0) = 1- p
- E(X) = p
- var(X) = p(1-p)
Binomial distribution
- Range up to n
- P(X = k) = C(n,k)p^k(1-p)^n-k for each k
- E(X) = np
- var(X) = np (1-p)
Poisson distribution?
- Unlimited range
- P(X = k) = (lambda)^k/ k! * e^ -(lambda)
- E(X) and var(X) = lambda
Difference eqution?
vk+1 = Avk
Solve difference equation?
vk = A^k v0
Markov process?
- Calculate prop for event a and event b using given info
- Place in Mx = b (each value matrix)
- For given time distribution is given by M^k x
Transition matrix?
Probability of moving from state to state as a vector
Markov process (Long term behavior)?
vk = v
- v is (lambda)-eigenvector of M
