Probability spaces

Question 1

Definition: The expected value of a random variable X defined on
a discrete probability space (Ω, F, P) is
E(X)

Answer 1

Definition: The expected value of a random variable X defined on
a discrete probability space (Ω, F, P) is
E(X) = X
ω∈Ω
X(ω) P(ω)
provided that this series converges absolutely (i.e., in any order).

The expected value of a random variable X with density function f
is
E(X) = Z ∞
−∞
tf (t)dt
provided that this integral exists

Question 2

Let X and Y be random variables defined on the same discrete

E(λX + µY

Answer 2

Let X and Y be random variables defined on the same discrete
probability space (Ω, F, P) and let λ, µ ∈ R.
Show that, if E(X) and E(Y ) exist then so does E(λX + µY ) and
E(λX + µY ) = λE(X) + µE(Y ).

Question 3

Definition: The variance of the random variable X is

Var(X) =

Answer 3

Definition: The variance of the random variable X is
Var(X) = E((X − E(X))^2) = E(X^2) − E(X)^2
whenever this expression exists.
The standard deviation of X is defined as square root ofVar(X).

Question 4

If c is any number, Var(cX)

Answer 4

If c is any number, Var(cX) = c^2 Var(X).

Question 5

Let X be a random variable. Show that the random
variable
X − E(X)
divided by
square rootVar(X)
has expected value 0 and variance 1

Answer 5

Let X be a random variable. Show that the random
variable
X − E(X)
divided by the square root of
Var(X)
has expected value 0 and variance 1

Question 6

For random variables X and Y we define the covariance of X and
Y

Answer 6

For random variables X and Y we define the covariance of X and
Y
Covar(X,Y ) = E( (X − E(X))( Y − E(Y )))

Question 7

Covar(X,Y )

Answer 7

Covar(X,Y ) = E(XY ) − E(X)E(Y

Question 8

Var(X + Y )

Answer 8

Var(X + Y ) = Var(X) + 2 Covar(X,Y ) + Var(Y

Question 9

Covar(X + a,Y + b) = Covar(X,Y ).

Answer 9

random variables X,Y and numbers
a, b ∈ R we have
Covar(X + a,Y + b) = Covar(X,Y ).

Question 10

t Covar(−, −) is a bilinear function,

Answer 10

Covar(−, −) is a bilinear function, i.e., for
random variables X,Y , Z and numbers λ, µ ∈ R we have
Covar(λX + µY , Z) = λ Covar(X, Z) + µ Covar(Y , Z)
and
Covar(X, λY + µZ) = λ Covar(X,Y ) + µ Covar(X, Z).

Question 11

correlation

Answer 11

The correlation between the random variables X and
Y is
ρ(X,Y ) = Covar(X,Y )/√ Var(X) Var(Y )

Question 12

ρ(a(X − b), c(Y − d))

Answer 12

for random variables X and Y and any
a, b, c, d ∈ R with a, c > 0 we have
ρ(a(X − b), c(Y − d)) = ρ(X,Y ).

Question 13

Var(X + Y )

Answer 13

For any two random variables X and Y
Var(X + Y ) = Var(X) + Var(Y ) + 2 Covar(X,Y )
= Var(X) + Var(Y ) + 2ρ(X,Y )√Var(X) Var(Y