Common Probability Distributions Flashcards Preview

CFA Level 1 - Quantitative Methods > Common Probability Distributions > Flashcards

Flashcards in Common Probability Distributions Deck (27)
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1
Q

Describes the probabilities of all possible outcomes for a random variable

A

Probability Distribution

2
Q

The number of possible outcomes that can be counted , and for each possible outcome, there is a measureable and positive probability

A

Discrete Random Variable

3
Q

A variable which the number of possible outcomes is infinite, even if lower and upper bounds exist.

A

Continuous Random Variable

4
Q

A function that defines the probability that a random variable, X, takes on a value equal to or less than a specific value, X.

A

Cumulative Distribution Function

*F(x) == P(X<=x)

5
Q

A variable for which the probabilities of all possible outcomes for a discrete random variable are equal

A

Discrete Uniform Random Variable

6
Q

The number of “successes” in a given number of trials, where the outcome is either a success or a failure. (The trials must be independent.)

A

Binomial Random Variable

7
Q

Calculation for the probability of x successes in n trials (in a Bernoulli random variable)

A

p(x) = (n!)/((n-x)!x! * (p^x*(1-p)^(n-x))

8
Q

Expected Value for Bernoulli Random Variable

A

E(x) = n * p

9
Q

Variance of a Binomial Random Variable

A

Var(X) = (n*p) * (1-p)

10
Q

A distribution defined over a range that spans between a lower limit, a, and upper limit, b, which serve as parameters of the distribution.

A

Continuous Uniform Distribution

11
Q

Properties of Continuous Uniform Distribution

A

*a<= x1<= x2<= b
**P(x<a> b) = 0
*** P(x1 < x < x2) = (x2-x1) / (b-a)
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12
Q

Properties of a Normal Distribution

A
  • X is normally distributed with mean,u, and variance, sigma^2
    • Skewness = 0, Mean = Median = Mode
  • ** Kurtosis = 3
  • *** A linear combination of normally distributed random variables is also normally distributed.
  • ** The probabilities of outcomes further above and below the mean get smaller and smaller, but do not go to zero.
13
Q

A distribution the specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent on the behavior of the others.

A

Multivariate Distribution

14
Q

Confidence Interval Ranges

A

90% C.I. is x +/- 1.65 sd
95% C.I. is x +/- 1.96 sd
99% C.I. is x +/- 2.58 sd

15
Q

A normal distribution that has been standardized so that it has a u = 0, and sigma = 1.

A

Standard Normal Distribution

* z = (x - u) /sigma

16
Q

The number of standard deviations above or below the mean

A

z-score

17
Q

The probability that a portfolio value or return will fall below a particular value or return over a given period of time.

A

Shortfall Risk

18
Q

The optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level.

A

Roy’s Safety First Ratio

* min P(Rp -Rl)

19
Q

A distribution that is generated by the function e^x, where x is normally distributed

A

Lognormal Distribution

  • Skewed to the right
    • Bounded from below by zero, so that it is useful for modeling asset prices which never take a negative value
20
Q

Calculation for Continuously Compounded Returns

A

Rcc = e^Rcc - 1

21
Q

Applications of Rcc

A
  • ln(EAR) = Rcc
    • Rcc = ln(S1/S0) = ln( 1 + HPR)
  • ** HPRt = e^Rcc*T - 1
22
Q

A technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values.

A

Monte Carlo Simulation

23
Q

Monte Carlo Simulations are used to:

A
  • Value complex securities
    • Simulate the profits/losses for a trading strategy
  • ** Calculate estimates of value at risk(VAR) to determine the riskiness of a portfolio of assets and liabilities
  • *** Simulate pension funds assets and liabilities overtime to examine the variability of the difference
  • ** Value portfolios of assets that have non-normal returns distributions
24
Q

Limitations of Monte Carlo Simulations

A
  • Fairly Complex
    • Will provide answers that are no better than the assumptions about the distributions of the risk factors and the pricing/valuation model that is used.
  • ** Not an analytic method, only statistical method.
25
Q

A simulation based on actual changes in value or risk factors over some prior period.

A

Historical simulation

26
Q

Advantage to historical simulations

A

distribution of changes in risk factors don’t have to be estimated.

27
Q

Disadvantages of historical simulations

A
  • risk factors of past may not be good predictors of the future
    • Cannot address “what-if” scenarios like MC simulations can. (i.e. What if we increase variance of risk B by 20%. How much does portfolio change? and in what direction?