Common Probability Distributions

Question 1

Name the seven probability distributions

Answer 1

• uniform
• binomial
• normal
• lognormal
• student’s t
• chi-square
• F-distribution

Question 2

Discrete Random Variable

Answer 2

• all possible outcomes can be listed

- ex. quoted stock price

Question 3

Continuous Random Variable

Answer 3

• where the number of points between the lower and upper bounds are infinite
• ex. the (P) of any single value of X=3

Question 4

Probability Distributions (2)

Answer 4

1. Probability function

2. Probability density function

Question 5

Probability function

Answer 5

• specifies the probability that a random variable takes on a specific value
• p(x) = P(X=x), where the capital X represents the random variable and the lowercase x represents a specific value that the random variable may take
• used for discrete random variables

Question 6

Probability Density Function

Answer 6

• denoted by f(x)
• used for continuous random variables
• denoted as: F(x) = P(X<=x)
• ie the sum of probabilities for the outcomes up to and including a specified outcome

Question 7

Continuous Uniform Distribution

Answer 7

• is defined over a range from a lower limit ‘a’ to an upper limit ‘b’. these limits serve as parameters of the distribution

=P(x1 <= X <= x2) = x2 - x1 / b - a

Question 8

X is a uniformly distributed continuous random variable between 10 and 20. Calculate the probability that X will fall between 12 and 18

Answer 8

=P(x1 <= X <= x2) = x2 - x1 / b - a

= 18 - 12 / 20 - 10
=.60

Question 9

An analyst predicts that the price per ounce of gold three years from now will be between $1500 and $1700. Assume gold prices follow a continuous uniform distribution. What is the probability that the price will be less than $1600 three years from now?

Answer 9

=P(x1 <= X <= x2) = x2 - x1 / b - a

1600 - 1500 / 1700 - 1500
= 50%

Question 10

Binomial Distribution

Answer 10

• the number of successes in a given number of Bernoulli trials

Question 11

Bernoulli trial

Answer 11

• there are only two possible outcomes, success or failure
• P(Y = 1) = p
• P(Y = 0) = 1 - p

Question 12

Binomial Random Variable Probability distribution formula

Answer 12

• probability of x successes in n trials

=P(x)=P(X=x) = nCx * p^x*(1-p)^n-x

p = the probability of success on each trial

ex.
n=10, P(x=7), p=.5

10C7 * .5^7(1-.5)^10-7= .117

Question 12

Binomial Random Variable Probability distribution formula

Answer 12

• probability of x successes in n trials

=P(x)=P(X=x) = nCx * p^x*(1-p)^n-x

p = the probability of success on each trial

ex.
n=10, P(x=7), p=.5

10C7 * .5^7(1-.5)^10-7= .117

Question 13

Mean and Variance of a Binomial Variable

Answer 13

Random Variable Mean Variance
Bernoulli, B(1,p) p p(1-p)
Binomial, B(n,p) np np(1-p)

ex Binomial B(10 coin flips, .5) (10.5)=5 5.5*.5

Question 14

Normal Distribution:
% of observations from mean
- 1 stdv
- 2 stdv
- 3 stdv

Answer 14

1 stdv: 68% of all observations fall within the interval of m+- 1 stdv

2 stdv: 95% of all observations fall within the interval of m+- 2 stdv

3 stdv: 99% of all observations fall within the interval of m+- 3 stdv

Question 15

Confidence intervals

- the three main ones

Answer 15

• 90% of all observations are in the interval of m+- 1.65 stdv
• 95% of all observations are in the interval of m+- 1.96 stdv
• 99% of all observations are in the interval of m+- 2.58 stdv

Question 16

Formula for standardizing a random variable X

Answer 16

= Z = (X - population mean, mue) / population stdv

Question 17

Pairwise return correlations formula

Answer 17

= pairwise = (n(n-1))/2

Question 18

Shortfall risk

Answer 18

• the risk that portfolio’s return will fall below a specified minimum level of return over a given period of time
• theshold level: min accepted ratio

Question 19

Safety first ratio

Answer 19

• used to measure shortfall risk
  = SF ratio = [Rp - RL] / stdv of portfolio
  *a higher ratio is safer

Rp = expected portfolio return
RL = threshold level

Question 20

Lognormal distribution

Answer 20

• completely defined by the mean and variance
• suitable for modeling asset prices
• cant be negative, starts at 0
• ## positive skew

Question 21

EAR for continuous compounding rates of return

Answer 21

EAR = e^r - 1

r = rate of return

Question 22

If we are given the holding period return over any time period, we can calculate the equivalent continuously compounded rate of return for that period with this formula:

Answer 22

= r = ln(HPR + 1)

Question 23

Student’s t-distribution

Answer 23

• symmetrical, bell-shaped and similar to a normal distribution
• has a lower peak and fatter tails (tails extend farther than normal)
• defined by degrees of freedom (df = n - 1)
• as the df increase, the t-distribution approaches the normal distribution

Question 24

Chi-Square distribution

Answer 24

• asymmetrical and defined by df
• cant be negative
• as df increase, looks more similar to a bell curve
• tests variance of a normally distributed population

Question 25

F distribution

Answer 25

• asymmetrical
• defined by two parameters: df1 and df2
• as both df1 and df2 increase, look more like a bell curve
• tests of equality of variances of two normally distributed populations from 2 independent random samples