SU2 - The Properties of Random Variables, The Normal and Its Related Distributions

Question 1

What is expected value of a random variable?

Answer 1

It is the weighted average of all possible values of X aka mean

Question 2

What is the Expectation of Sums?

E(X1+X2)

Answer 2

E(X1)+E(X2)

Question 3

What is the result of E(c1X1+c2X2)?

Answer 3

E(c1X1) + E(c2X2) = c1E(X1) + c2E(X2)

Question 4

What is the expectation of a binomial distribution?

Answer 4

np

Question 5

What is Jensen’s inequality?

Answer 5

g(E(X)) ≤ E(g(X))

Question 6

What is the result of E (aX + b)?

Answer 6

E (aX + b) = aE(X) + b

Question 7

What is the expectation of a Bernoulli distribution?

Answer 7

p

Question 8

Is the expectation of the function of a random variable equals to the function of the random variable’s expectation?
E [g (X)] = g (E [X])

Answer 8

No, unless g(X) is a linear function, Jensen’s inequality

Question 9

Is the expectation of the product of random variables equal to the product of their expectations?

Answer 9

No, only if the random variables with the product are independent

Question 10

What is the Cauchy distribution?

Answer 10

Student t distribution with 1 degree of freedom. In this case, the expectation may not exist

Question 11

What is the variance?

Answer 11

To measure how “spread out” the values of a random variable are
𝑉𝑎𝑟𝑋=𝐸[(𝑋−𝜇)2] or 𝐸(𝑋2)−𝜇2

Question 12

For any constant c, what is the value of Var(c)?

Answer 12

0. For any constant, value of variance is 0 because there is no variance at all.

Question 13

For any constants a and b, Var(aX+b) = ??

Answer 13

For any constants a and b, Var(aX+b) = a^2 Var(X)

When we add b to X, we only shift the distribution of X laterally. The shape of the distribution of X stays the same; therefore, the variance of X remains unchanged

Question 14

For any constants a and b, and sd ( aX+b ) = ??

Answer 14

|a| sd (X)

Question 15

What does standardization mean?

Answer 15

To re-centre the expectation of the random variable to 0 and to normalise its variance to 1, i.e. E(X) = 0 and Var(Z) = 1.

Question 16

If 𝑋1 and 𝑋2 are uncorrelated/independent, Var𝑋1+𝑋2 =?

Answer 16

Var(X1) + Var(X2)

Question 17

Is the variance of the sum of random variables always equal to the sum of their variances?

Answer 17

No, unless the variables are independent

Question 18

What happens to the covariance when X and Y are independent?

Answer 18

If X and Y are independent, then Cov(X,Y) = 0, and E(XY) = E(X)E(Y)

Question 19

What is the correlation coefficient?

Answer 19

The correlation coefficient can be thought of as a standardised covariance that does not depend on units of measurement.

corr(X,Y) = 1(-1) means a perfect positive (negative) linear relationship

Question 20

What is the variance of a bernoulli distribution?

Answer 20

Var(X) = np(1-p)

Question 21

What is the alternate expression for Cov(X, Y)?

Answer 21

E(XY) - 𝜇𝑋𝜇𝑌

E(XY) - E(X)E(Y)

Question 22

What are the three properties of covariances?

Answer 22

• For any constant a, Cov(a,X) = 0.
• For random variable Z, Cov(X+Y,Z) = Cov(X,Z)+Cov(Y,Z).
• For any constants a1 and a2, Cov(a1 X,a2 Y) = a1 a2 Cov(X,Y).

Question 23

Does zero correlation imply independence?

Answer 23

Yes

Question 24

What happens if you add a constant to a random variable in a covariance?

Answer 24

Does not affect its covariance with another random variable

E(X) = E(E(X│Y)) = E(c) = c

Question 25

What happens if you multiply a constant to a random variable in a covariance?

Answer 25

Affect the covariance by the same multiple

Question 26

What is a conditional expectation?

Answer 26

Explaining one variable in terms of the other variable

Question 27

What is the law of Iterated Expectation?

Answer 27

E[E(Y|X)] = E(Y)

Question 28

How to write a conditional expectation of a random variable Y given a random variable X?

Answer 28

E(Y|X)

Question 29

E(X|X) = ?

Answer 29

X

Question 30

E(c(X)|X) = ?

Answer 30

c(X)

Question 31

if X and Y are independent, E(Y|X) = ?

Answer 31

E(Y)

Question 32

if E(Y|X) = E(Y), then Cov(X,Y) = ?

Answer 32

0

Question 33

E(XY) can directly jump to E(E(XY|Y)?

Answer 33

Yes

Question 34

What is a normal random variable?

Answer 34

The normal random variable is symmetrically distributed around its mean μ. The peak of the normal density function occurs at Y=μ and the density function is concentrated around μ.

Question 35

How to read this? | X ∼ N(µ, σ2)

Answer 35

The normal distribution with mean µ and variance σ2

Question 36

If X ∼ N(µ, σ2) and if Y = aX+b and a =/= 0, then y = ?

Answer 36

Y has a normal distribution with mean aµ+b and variance a2o2 every linear function of X will also have a normal distribution.

Question 37

What is the standard normal?

Answer 37

A normal random variable with mean 0 and variance 1.

Question 38

What is the Chi-Square mean and variance?

Answer 38

``` E(Y) = m Var(Y) = 2m ```

Question 39

How is the chi-square related to the standard normal?

Answer 39

When the degrees of freedom in a Student-t goes to infinity, the Student-t becomes the standard normal.