continuous random var

Question 1

gaussian integral

Answer 1

integral from -inf to +inf of e^(-x^2) = sqrt(PI)

Question 2

p.d.f of standard normal distribution of random variable Z

Answer 2

fz(z) = 1/(sqrt(2PI))e^(-z^2/2)

Question 3

expectation and variance of standard normal distribution

Answer 3

E(Z) = 0
Var(Z) = 1

Question 4

the cumulative distribution function of N(0,1)

Answer 4

Fz(z) = integral from -inf to z of 1/(sqrt(2pi))e^(-t^2/2)dt

the notation Φ is commonly used to represent c.d.f and φ to represent p.d.f

there is a relationship between Φ(z) and Φ(-z)
Φ(-z) = 1 - Φ(z)

Question 5

p.d.f of random variable X which has normal distribution with mean a and variance b^2

Answer 5

fx(x) = 1/(sqrt(2pib^2))e^(-(x-a)^2/2b^2)

Question 6

define general normal variable X in terms of normal random variable Z

Answer 6

X = a + b*Z,
E(X) = a
Var(X) = b^2
we refer to transforming X to get a standard N(0,1) random variable as standardising X

Question 7

normal distribution in Matlab

Answer 7

p.d.f. - normpdf(x,mu, sigma)
c.d.f. - normcdf(x,mu, sigma)
invert c.d.f. to find alfa quantile: norming(alfa, mu, sigma)

Question 8

the mean of vector X = (X1, X2, …, Xn)^T

Answer 8

a = (E(X1), E(X2),…,E(Xn))

Question 9

P.D.F of random variable X with mean a and covariance matrix Σ which has a multivariate normal distribution is

Answer 9

fx(x1, x2, …, xn) = 1/(sqrt((2pi)^k|Σ|))e^(-(X-a)^TΣ^-1*(X-a)/2)
where k is a real k-dimension column vector and |Σ| is the determinant of Σ