Probability Distributions

Question 1

Probability distribution

Answer 1

A probability distribution lists all the possible outcomes of an experiment, along with their associated probabilities.

Question 2

Discrete Random Variable

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A Discrete Random Variable has positive probabilities associated with a finite number of outcomes.

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Continuous Random Variable

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A continuous random variable has positive probabilities associated with a range of outcome values–the probability of any single value is zero.

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Set of possible outcomes of a specific discrete random variable

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Finite set of values (in a discrete distribution, p(x)=0 if it cannot happened, and p(x)>0 if it can).

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Probability Function

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P(X=x) = p(x), such that 0<=1 and Σp(x)=1

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Probability Density Function (pdf)

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Function for a continuous random variable used to determine the probability it will fall in a particular range

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Cumulative Distribution Function (cdf)

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Discrete Uniform Distribution

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Distribution where there are n discrete, equally likely outcomes.

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Binomial Distribution

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Probability distribution for a binomial (discrete) random variable that has two possible outcomes.

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Probability of an outcome under a discrete uniform distribution

Answer 10

1/n

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Probability of an outcome under binomial distribution (with p = probability of success)

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Binomial tree

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Illustrates the probabilities of all the possible values that a varaible can take on given the probability of an up-move and the magnitude of an up-move

Question 13

Continuous Uniform Distribution

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The probability of X occuring in a possible range is the length of the range relative to the total of all possible values. If a and b are the limits, then:

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Normal Probability distribution

Answer 14

1) it is symmetrical and bell-shaped with a peak in the center
2) mean = median = mode,
3) the normal distribution is defined by the

mean and standard deviation; skew = 0; kurtosis = 3

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Multivariate Distribution

Answer 15

Describes the probabilities for more than one random variable

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Univariate Distribution

Answer 16

Describes the probabilities for a single random variable

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Confidence interval

Answer 17

The range within which we have a given level of confidence of finding a point estimate

Question 18

90% confidence interval

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μ +/- 1.65 standard deviations

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95% confidence interval

Answer 19

μ +/- 1.96 standard deviations

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99% confidence interval

Answer 20

μ +/- 2.58 standard deviations

Question 21

Probability that a normally distributed random variable X will be within A standard deviations of its mean

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Twice the cumulative left-hand tail probability F(-A), where F(A) is the cumulative standard normal probability of A

Question 22

z-table

Answer 22

Used to find the probability that X will be less than or equal to a given value

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P(X<>

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P(X>x)

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Shortfall risk

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The probability that a portfolio's value (or return) will fall below a specific value over a given time. Greater safety-first ratios indicate a smaller shortfall risk.

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Roy's safety-first criterion

Answer 26

The optimal portfolio minimizes shortfall risk.

Question 27

Lognormal distribution

Answer 27

If x is normally distributed, e^x follows a lognormal distribution. A lognormal distribution is often used to model asset prices, since it cannot be negative and can take on any positive value.

Question 28

The continuously compounded rate via HPR

Answer 28

ln ( 1 + HPR )

Question 29

Monte Carlo simulation

Answer 29

Use of randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values. It is complex and will provide answers no better than assumptions used.

Question 30

Historical simulation

Answer 30

Randomly selected past values to generate a distribution of possible security values. It cannot consider the effects of significant events that did not occur in a sample period.

Question 31

Standardization of a random variable z =

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Sharpe Ratio

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Treynor Ratio

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Probability Function

Answer 34

It specifies the probability that a random variable is equal to a specific value In other words it is the probability that random variable X takes on value x or p(x) = P(X = x )

Question 35

Shortfall Risk

Answer 35

Shortfall Risk is the probability that a portfolio value or return will fall below a particular target value or return over given time period

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Variance in terms of Expected Value

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Safety-first ratio for portfolio P

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S

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Properties of Cummulative Distribution Function

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Properties of Univariate or Multivariate Distributions

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Standard Deviation other Formula

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Standard Deviation other Formula

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How to find negative Z values

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