Definitions (with Equations)

Question 1

Define a cumulative distribution function and its properties (PMF)

Answer 1

For discrete distributions:

FX (x) = P({w : X(w) ≤ x}) = P(X ≤ x) = X
SUM ∀xi≤x pX (xi).

Properties:

1) 0 ≤ FX (x) ≤ 1.
(2) limx→−∞ FX (x) = 0.
(3) limx→+∞ FX (x) = 1.
(4) Non-decreasing function.

Question 2

Define an expectation for a discrete random variable X

Answer 2

Expected value, mean value of a discrete random variable

μX = E[X] =
nX
k=1
xk · pX (xk).

Question 3

Define R’th moment of a discrete random variable X

Answer 3

For a discrete random variable X:

μr = E[Xr ] =
nX
k=1
xr
k · pX (xk)

Question 4

Define R’th centred moment of a discrete random variable X

Answer 4

For a discrete random variable X:

μr = E[(X − μX )r ] =
nX
k=1
(xk − mX )r · pX (xk)

Question 5

Define a bernoulli variable

Answer 5

X ~ Be(p), has outcomes 0 and 1with the following probabilities:

pX (1) = p and pX (0) = 1 − p

Question 6

Define probability density function (PDF)

Answer 6

Probability distribution for function (CDF) for a continuous variable:

X (x) = P(x < X ≤ x + dx)

Question 7

Define expectation for a continuous random variable X

Answer 7

X = E[X] =
Z
DX
x · fX (x)dx

Question 8

Define R’th moment of a continuous random variable X

Answer 8

μr = E[Xr ] =
Z
DX
xr · fX (x)dx

Question 9

Define R’th centred moment of a continuous random variable X

Answer 9

r = E[(X − μX )r ] =
Z
DX
(x − mX )r · fX (x)dx

Question 10

Define Skewness of a continuous random variable

Answer 10

“ X − μX
σX
3#
= E[(X − μX )3]
σ3
X

Question 11

Define normal distribution

Answer 11

(x) = 1
σ√2π exp − 1
2
x − μ
σ
2!
, −∞ < x < ∞

Question 12

Define a lognormal variable

Answer 12

X ∼ LN (λ, φ) ⇔ ln(X) ∼ N (μ, σ2).

Its PDF reads

(x) = 1
φx√2π exp − 1
2
ln(x) − λ
φ
2!
, x ≥ 0

Question 13

Define a two dimensional random variable, Z

Answer 13

Z=(X)
(Y)

is a set of two variable X and Y, characterised by its joint CDF:

Z = FX,Y (x, y) = P({X ≤ x} ∩ {Y ≤ y})

The PDF is given as

(x, y) dxdy = P({x < X < x + dx} ∩ {y < Y < y + dy}

Question 14

Define Marginal Distribution

Answer 14

Marginal distrbution of each component is obtained by integrating the joint distribution over the other variable

X (x) =
Z
DY
fX,Y (x, y)dy, and fY (y) =
Z
DX
fX,Y (x, y)dx

Question 15

Define expectation for a two dimensional random variable X

Answer 15

[g(X, Y )] =
Z
DZ
g(x, y)fZ (x, y) dxdy

The individual mean values would be given as

E[X] = μX =
Z
DZ
x · fZ (x, y) dxdy and E[Y ] = μY =
Z
DZ
y · fZ (x, y) dydx.

Question 16

Define covariance

Answer 16

Cov[X, Y ] = E[(X − μX )(Y − μY )] =
Z
DZ
(x − μx)(y − μy )fZ (x, y) dxdy

Cov[X, Y ] = E[xy] − E[x]E[y]