(R9) Common Probability Distributions

Question 1

Probability distribution

Answer 1

Lists all the possible outcomes of an experiment, along with their associated probabilities. A probability distribution completely describes a random variable.

AKA: probability function

Question 2

Discrete random variable

Answer 2

Has positive probabilities associated with a finite number of outcomes (countable, non zero probabilities for each outcome)

Question 3

Continuous random variable

Answer 3

Has positive probabilities associated with a range of outcome values. The probability of any single value is zero (Uncountable). Described by probability density function instead of a probability function or probability distribution

Question 4

Probability Function

Answer 4

All possible outcomes plus associated probabilities. Completely describes the variable

Question 5

Cumulative Distribution Function

Answer 5

A function giving the probability that a random variable is less than or equal to a specified value. Either increases or remains constant over each possible outcome (Used for discrete and continuous random variables)

Question 6

Discrete Uniform Distribution and 3 characteristics

Answer 6

The probability of every finite possible outcome is equally likely.

1. Outcomes are countable
2. probability between 0 and 1
3. Sum of all probabilities equal 1

Question 7

Binomial Random Variable

Answer 7

The number of successes in n Bernoulli trials for which the probability of success is constant for all trials and trials are independent

Question 8

Bernoulli random Variable

Answer 8

A random variable having outcomes 0 and 1

Question 9

Bernoulli trial

Answer 9

An experiment that can produce one of two outcomes (success or failure)

Question 10

Probability of bernoulli trial

Answer 10

1/n where n is the number of trials

Question 11

Probability of binomial distribution

Answer 11

Question 12

Mean and variance formula for Bernoulli random variable and binomial random variable

Answer 12

Bernoulli Mean = p

Bernoulli Variance = p (1 - p)

Binomial Mean = np

Binomial Variance = np (1-p)

Question 13

Binomial Tree

Answer 13

The graphical representation of a model of asset price dynamics in which at each period, the asset moves up with probability p or down with probability (1-p)

Question 14

Continuous Uniform Distribution

P(X₁ <= X <= X₂)

Answer 14

Described by a lower limit a and an upper limit b (a and b are parameters of the distribution)

Probability = X₂ - X₁ / b - a

Question 15

Normal Distribution

Answer 15

Completely described by mean and variance with a skew of 0 and kurtosis of 3. This distribution is only used for continuous random variables

Question 16

Central Limit Theorem

Answer 16

The sum of a large number of independent random variables are approximatelly normally dsitributed

Question 17

Univariate Normal Distribution

Answer 17

A distribution that specifies the probabilities for a single random variable

Question 18

Multivariate Normal Distribution

Answer 18

A probability distribution for a group of random variables that is completely defined by the means and variances of the variables plus all the correlations between pairs of variables

Question 19

Confidence Intervals (90, 95, and 99%)

Answer 19

90%: sample mean + or - 1.65s

95%: sample mean + or - 1.96s

99%: sample mean + or - 2.58s

Question 20

Standardizing definition and formula

Answer 20

Refers to the number of standard deviations away from the mean an observation lies

z = (observed value - mean) divdided by standard deviation

Question 21

Roy’s safety first criterion (Safety first ratio) formula

Answer 21

(Expected return on portfolio - return on risk-free return) divided by standard deviation

The larger the ratio the better

Question 22

Shortfall Risk

Answer 22

The risk that portfolio value will fall below some minimum acceptable level over some time horizon

Question 23

Define lognormal distribution and list 3 characteristics

Answer 23

A random variable x follows a lognormal distribution if its natural log (LN Y) is normally distributed. Used to model asset prices, while normal distribution model returns.

1. Bounded by zero on the lower end
2. Upper end is unbounded
3. Positvely skewed (skewed to right)

Question 24

Formula for lognormal distribution

Answer 24

Lognormal distribution = (VT/V0), where VT is ending value asset price and V0 is beginning value asset price

Normal distribution = Ln (VT/V0)

Question 25

Continuously Compounded Return Formula

Answer 25

The natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price

ln(1+HPR) or ln(Vt/Vo)