Week 14 - Rules of probability, and probability distributions Pt2

Question 1

What is a random variable?

Answer 1

a numerical description of the outcome of an experiment.

Question 2

What is a discrete random variable?

Answer 2

may assume either a finite number of values or an infinite sequence of values.

The probabilities associated with each possible value are typically represented in a probability distribution (e.g., binomial distribution).

Question 3

Examples of a discrete random variable

Answer 3

The number of heads when flipping a coin 3 times. The possible outcomes are 0, 1, 2, or 3 heads.

The number of students in a classroom (which can be a whole number: 1, 2, 3, etc.).

Question 4

What is a continuous random variable?

Answer 4

may assume any numerical value in an interval or collection of intervals.

can take on an infinite number of possible values within a certain range.

Question 5

Example of a continuous random variable

Answer 5

The height of a person can be any value within a certain range (e.g., between 4 feet and 7 feet).

The time taken for a runner to complete a race, which can take any value in a continuous range (e.g., 10.2 seconds, 10.25 seconds, etc.).

Question 6

What does the probability for a random variable describe?

Answer 6

describes how probabilities are distributed over the values of the random variable.

Question 7

What is the probability distribution defined by?

Answer 7

is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable.

Question 8

What are the required conditions for a discrete probability function?

Answer 8

f(x)≥0
∑f(x) = 1

Question 9

How to put discrete probability distributions on a graph

Answer 9

x axis - values of random variable x
y axis - probability

Question 10

What is the expected value or mean of a random variable?

Answer 10

a measure of its central location.
𝐸(𝑋)= 𝜇= ∑𝑥𝑝(𝑥)

E(X) or 𝜇 - the expected value (mean) of the random variable X
x - each possible value of the random variable
p(x) - the probability of x occurring

Question 11

What does the variance summarise?

Answer 11

the variability in the value of a random variable
𝑉𝑎𝑟(𝑋)= 𝜎^2= ∑(𝑥−𝜇)ˆ2 𝑝(𝑥)
𝑉𝑎𝑟(𝑋)= 𝜎^2= 𝐸(𝑋−𝐸(𝑋))ˆ2
= 𝐸(𝑋ˆ2 ) - [𝐸(𝑋)]ˆ2

E(X) or 𝜇 - the expected value (mean) of the random variable X
x - each possible value of the random variable
p(x) - the probability of x occurring

The standard deviation, 𝜎, is defined as the positive square root of the variance.

Question 12

4 important properties

Answer 12

1. 𝐸(𝑎𝑋)=𝑎𝐸(𝑋)
2. 𝐸(𝑎𝑋+𝑏)=𝑎𝐸(𝑋)+𝑏
3. 𝑉𝑎𝑟(𝑎𝑋)=𝑎^2 𝑉𝑎𝑟(𝑋)
4. 𝑉𝑎𝑟(𝑎𝑋+𝑏)=𝑎^2 𝑉𝑎𝑟(𝑋)

Question 13

Example:
A random variable X takes the value of 0 with probability of 0.5, 1 with probability of 0.3, and 2 with a probability of 0.2

Calculate E(X), E(3X+3), E(Xˆ2 ), Var(X), Var(3x), and Var(3x+3)

Answer 13

X = 0 with the probability of 0.5
X = 1 with the probability of 0.3
X = 2 with the probability of 0.2

𝐸(𝑋)= 𝜇= ∑𝑥𝑝(𝑥)
E(X) = 0x0.5 + 1x0.3 + 2x0.2
E(X) = 0.7

E(3X+3) = 3E(X)+3
E(3X+3) = 3x0.7 +3 = 5.1

E(Xˆ2) = ∑𝑥ˆ2𝑝(𝑥)
E(Xˆ2) = 0ˆ2 x 0.5 + 1ˆ2 x 0.3 + 2ˆ2 x 0.2 = 1.1

𝑉𝑎𝑟(𝑋)=𝐸(𝑋ˆ2 ) - [𝐸(𝑋)]ˆ2
Var(X) = 1.1 - (0.7)ˆ2
Var(X) = 0.61

Var(3X)=3ˆ2Var(X)
Var(3X) = 9(0.61)
Var(3X) = 5.49

Var(3X+3) = 3ˆ2Var(X)
Var(3X+3) = 5.49

Question 14

What is covariance?

Answer 14

measures how two random variables vary together

Question 15

What are 3 expressions for covariance?

Answer 15

𝐶𝑜𝑣(𝑋,𝑌)=𝐸[(𝑋 − µ_𝑋)(𝑌 − µ_𝑌)]

𝐶𝑜𝑣(𝑋,𝑌)=𝐸(𝑋𝑌) − 𝐸(𝑋)µ_𝑌 − µ_𝑋𝐸(𝑌) + µ_𝑋 µ_𝑌

𝐶𝑜𝑣(𝑋,𝑌)= 𝐸(𝑋𝑌) − 𝐸(𝑋)𝐸(𝑌)

µ_X -
µ_Y - E(X)

Question 16

Expected value and variance
E(X+Y) =
Var(X+Y) =

Answer 16

E(X+Y) = E(X) + E(Y)

Var(X+Y) = aˆ2Var(X) + bˆ2Var(Y) + 2abCov(X,Y)
= aˆ2Var(X) + bˆ2Var(Y) + 2abρ_(𝑋,𝑌)σ_𝑋 σ_𝑌

Question 17

Expected value and variance
E(aX + bY) =
Var(aX+bY) =

Answer 17

E(aX + bY) = aE(X) + bE(Y)
Var(aX+bY) = aˆ2Var(X) + bˆ2Var(Y) + 2abCov(X,Y)
= aˆ2Var(X) + bˆ2Var(Y) + 2abρ_(𝑋,𝑌) σ_𝑋 σ_𝑌

Question 18

Portfolio risk and return (N=2)

Answer 18

Suppose we have 2 assets, and we invest a fraction a1 in asset 1, and a2 in asset 2, then
a_1 + a_2 = 1

The mean return on the portfolio is the weighted average of the mean returns on each of the assets:
μ_p = a_1μ_1 + a_2μ_2

The variance of the portfolio is the weighted sum of the covariances:
σˆ2 = a_1 σ_11 + 2a_1a_2 σ_12 + a_2ˆ2 σˆ22
= aˆ2 σ_1 ˆ2 + 2a_1 a_2 p_12 σ_1σ_2 + a_2 ˆ2 σ_2 ˆ2

Question 19

What function is the mean of the composition of the portfolio?

Answer 19

a linear function of the composition of the portfolio

Question 20

What function is standard deviation?

Answer 20

a convex function

Question 21

Why does diversification occur?

Answer 21

Diversification occurs because combining assets that are not perfectly correlated reduces overall risk.

Even if each individual asset is risky, their combined risk can be lower if their returns move differently.

the riskiness of a portfolio is less than the average of its components

Question 22

What is the diversification effect?

Answer 22

σP <= [ω σA + (1- ω) σB], 0 < ω <= 1

Question 23

When p = 1 what is the diversification effect?

Answer 23

no diversification effect obtains

Question 24

When p < 1 what is the diversification effect?

Answer 24

diversification effect obtains

Question 25

What does p need to be for the diversification effect to obtain?

Answer 25

p need not be <0 for diversification effect to obtain p<1 is sufficient

Question 26

The higher or lower p is better the diversification?

Answer 26

lower the p, better the diversification

Question 27

When does σ_P = 0?

Answer 27

If and only if ρ = -1 and ω = σB/(σA + σB)

Question 28

What is the definition of a risk-free asset?

Answer 28

σF = 0 This definition ensures that E[RF] = RF

Question 29

Two perfectly negatively correlated and equally volatile securities with expected returns of 10% and 7% are combined into an investment portfolio. If the portfolio is to be as riskless as possible, its expected return should be equal to a) 8.5% b) 10% c) 7% d) 5%

Answer 29

Since these securities are perfectly negatively correlated, we can form a riskless portfolio.   Proportion to invest in A = 𝑤_𝐴=𝜎_𝐵/(𝜎_𝐴+𝜎_𝐵 ) Also, we are told that 𝜎_𝐴=𝜎_𝐵 Therefore, 𝑤_𝐴 = 0.5 and 𝑤_𝐵=0.5 [Note that 𝑤_𝐴+𝑤_𝐵=1].   Return of a portfolio with 50% invested in A and 50% in B = 0.5x10%+0.5x7% = 8.5%

Question 30

Two perfectly positively correlated securities are combined into a portfolio. One security is expected to return 10% with a volatility of 20% and the other is expected to return 15% with a volatility of 25%. If the portfolio is required to have a (target) return of 20%, what is its volatility? a) 20% b) 30% c) 22.5% d) 25%

Answer 30

20%= 𝑤_𝐴x𝑟_𝐴+𝑤_𝐵x𝑟_𝐵 = 𝑤_𝐴x𝑟_𝐴+(1−𝑤_𝐴)x𝑟_𝐵 20%= 𝑤_𝐴x10%+(1−𝑤_𝐴 )x15% 𝑤_𝐴= −1 and 𝑤_𝐵= 2 Volatility of a portfolio with 𝑤_𝐴= −1 (-100% invested in A) and 𝑤_𝐵= 2 (200% invested in B), with correlation between A and B = 1:   𝜎_𝑃^2= (𝑤_𝐴∗𝜎_𝐴+𝑤_𝐵∗𝜎_𝐵 )^2 𝜎_𝑃^2= (−1∗20%+2∗25%)^2 𝜎_𝑃^2= (30%)^2 𝜎_𝑝=30%