Common Probability Distributions Flashcards
Name the seven probability distributions
- uniform
- binomial
- normal
- lognormal
- student’s t
- chi-square
- F-distribution
Discrete Random Variable
- all possible outcomes can be listed
- ex. quoted stock price
Continuous Random Variable
- where the number of points between the lower and upper bounds are infinite
- ex. the (P) of any single value of X=3
Probability Distributions (2)
- Probability function
2. Probability density function
Probability function
- 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
Probability Density Function
- 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
Continuous Uniform Distribution
- 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
X is a uniformly distributed continuous random variable between 10 and 20. Calculate the probability that X will fall between 12 and 18
=P(x1 <= X <= x2) = x2 - x1 / b - a
= 18 - 12 / 20 - 10
=.60
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?
=P(x1 <= X <= x2) = x2 - x1 / b - a
1600 - 1500 / 1700 - 1500
= 50%
Binomial Distribution
- the number of successes in a given number of Bernoulli trials
Bernoulli trial
- there are only two possible outcomes, success or failure
- P(Y = 1) = p
- P(Y = 0) = 1 - p
Binomial Random Variable Probability distribution formula
- 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
Binomial Random Variable Probability distribution formula
- 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
Mean and Variance of a Binomial Variable
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
Normal Distribution: % of observations from mean - 1 stdv - 2 stdv - 3 stdv
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
Confidence intervals
- the three main ones
- 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
Formula for standardizing a random variable X
= Z = (X - population mean, mue) / population stdv
Pairwise return correlations formula
= pairwise = (n(n-1))/2
Shortfall risk
- 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
Safety first ratio
- used to measure shortfall risk
= SF ratio = [Rp - RL] / stdv of portfolio
*a higher ratio is safer
Rp = expected portfolio return RL = threshold level
Lognormal distribution
- completely defined by the mean and variance
- suitable for modeling asset prices
- cant be negative, starts at 0
- ## positive skew
EAR for continuous compounding rates of return
EAR = e^r - 1
r = rate of return
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:
= r = ln(HPR + 1)
Student’s t-distribution
- 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
Chi-Square distribution
- 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
F distribution
- 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
