Continuous Functions and Expectation

Question 1

What is the cumulative distribution function of a pdf?

Answer 1

(Integral from a to x) of f(u) {u=dummy variable}, where a is the lower limit of X

Question 2

If g[X] is a function of X then what is its expectation and what is its variance?

Answer 2

E(g[X]) = (integral) g[x]f(x) dx

Var(g[X])= (integral) (g[x])^2f(X) dx - [E(g[X])]^2

Question 3

E(aX+c)=?
Var(aX+c)=?

E(aX+bY)=?
Var(aX+bY)=?

E(X1 +/- X2 +/- X3…)=?
Var(X1 +/- X2 +/- X3…)=?

Answer 3

E(aX+c)= aE(X) + c
Var(aX+c)= a^2Var(X)

E(aX+bY)= aE(X) + bE(Y)
Var(aX+bY)= a^2Var(X)+b^2Var(Y)

E(X1 +/- X2 +/- X3…)= E(X1) +/- E(X2)..
Var(X1 +/- X2 +/- X3…)= Var(X1) + Var(X2)…

Question 4

What is mean, median and mode of a pdf?

Answer 4

Mean: (Integral) xf(x) dx
Median: F(m) = 0.5
Mode: f’(x)=0

Question 5

What makes any f(X) a valid probability density?

Answer 5

Integral between upper and lower limit= 1

f(x) > (or equal to) 0 for all x in the domain