UNIVARIATE CASE

TERMINOLOGY

NOTATION

RELATED FORMULAS

RANDOM VARIABLE

X, Y, Z, etc.

 

PARTICULAR VALUE OF A RANDOM VARIABLE

a, b, i, j, n, x, y, z, w etc.

 

CUMULATIVE DISTRIBUTION FUNCTION (cdf) OR DISTRIBUTION FUNCTION (df)

F(.)

F(b) = P(X²b)

P{a<X²b) = F(b) - F(a)

 

FOR DISCRETE CASE

F(a) =

FOR CONTINUOUS CASE

F(a) =

PROBABILITY MASS FUNCTION (pmf) (discrete case)

p(.)

p(a) = P(X=a)

PROBABILITY DENSITY FUNCTION (pdf) (continuous case)

f(.)

f(a) = F(a)

P(X=a) = 0

P(XëB) =

P{a²X²b) =

 

EXPECTATION OF A RANDOM VARIABLE

(EXPECTED VALUE OF X)

E[X]

FOR DISCRETE CASE E[X] =

 

FOR CONTINUOUS CASE

E[X] =

EXPECTATION OF A FUNCTION OF A RANDOM VARIABLE (g(X))

E[g(X)]

FOR DISCRETE CASE

E[g(X)] =

FOR CONTINUOUS CASE

E[g(X)] =

IN GENERAL

E[aX + b] = aE[X] + b

nth MOMENT OF A RANDOM VARIABLE

E[X]

FOR DISCRETE CASE

E[X] =

FOR CONTINUOUS CASE

E[X] =

 

VARIANCE OF A RANDOM VARIABLE

Var(X)

Var(X) = E[(X - E[X])] = E[X] - (E[X])

Var(aX + b) = aVar(X)

 

MOMENT GENERATING FUNCTION OF A RANDOM VARIABLE

f(t) or f

IN GENERAL

f(t) = E[e]

FOR DISCRETE CASE

f(t) =

FOR CONTINUOUS CASE

f(t) =

IN GENERAL

f'(0) = E[X]

f = E[X], n³1

 

 

 

 

 

 

 

 

MULTIVARIATE CASE

(JOINTLY DISTRIBUTED RANDOM VARIABLES)

TERMINOLOGY

NOTATION

RELATED FORMULAS

JOINT CUMULATIVE PROBABILITY DISTRIBUTION FUNCTION (jcdf or jdf) OF X AND Y

F(.,.)

F(a,b) = P{X²a,Y²b)

MARGINAL DISTRIBUTION FUNCTION OF X (Y)

F (F)

F= F(a,°) = P{X²a)

(F= F(°,b) = P{Y²b))

JOINT PROBABILITY MASS FUNCTION OF X AND Y (jpmf) (discrete case)

p(.,.)

p(x,y) = P{X=x,Y=y}

p=

 

p=

 

JOINT PROBABILITY DENSITY FUNCTION OF X AND Y (jpdf) (continuous case)

f(.,.)

P(XëA, YëB) = f(x,y)dxdy

f =

f =

 

EXPECTATION OF FUNCTIONS OF TWO VARIABLES X AND Y

E[g(X,Y)]

FOR DISCRETE CASE

E[g(X,Y)] = g(x,y)p(x,y)

FOR CONTINUOUS CASE

E[g(X,Y)] = g(x,y)f(x,y)dxdy

THE COVARIANCE OF X AND Y

Cov(X,Y)

IN GENERAL

Cov(X,Y) = E[(X - E[X])(Y - E[Y])]

              = E[XY] - E[X]E[Y]

Var(X + Y] = Var[X] + Var[Y] + 2Cov[X,Y]

Var[ X - Y] = Var[X] + Var[Y] - 2Cov[X,Y]

If X and Y are independent then Cov[X,Y] = 0