# Reading 10- part 1

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2018-02-18 17:35

## The Discrete Uniform Distribution

Question | Answer |
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definition | The distribution has a finite number of specified outcomes, and each outcome is equally likely. |

The cdf has two other characteristic properties | The cdf lies between 0 and 1 for any x: 0 ≤ F(x) ≤ 1. As we increase x, the cdf either increases or remains constant. |

The Binomial Distribution | When we make probability statements about a record of successes and failures, or about anything with binary outcomes, we often use the binomial distribution |

binomial random variable | X is defined as the number of successes in n Bernoulli trials. A binomial random variable is the sum of Bernoulli random variables Yi, i = 1, 2, …, n: |

Now we extend the model to describe stock price movement on three consecutive days. Each day is an independent trial. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 − p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. We graph stock price movement in Figure 2, where we now associate each of the n = 3 stock price moves with time indexed by t. The shape of the graph suggests why it is a called a binomial tree. Each boxed value from which successive moves or outcomes branch in the tree is called a node; in this example, a node is potential value for the stock price at a specified time.

## CONTINUOUS RANDOM VARIABLES

Question | Answer | |
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pdf for a uniform random variable is | f(x) = 1/(b-a) for x between a and b & f(x)= 0 otherwise | |

middle line of the expression for the cdf captures this relationship | f(x)=0 for x =< a, = (x-a)/(b-a) for x between a and b, = 1 for x >= b | |

mathematical operation that corresponds to finding the area under the curve of a pdf f(x) from a to b | the integral of f(x) from a to b | |

The Normal Distribution | The normal distribution is completely described by two parameters—its mean, μ, and variance, σ2 | normal distribution has a skewness of 0 (it is symmetric),The normal distribution has a kurtosis (measure of peakedness) of 3; its excess kurtosis (kurtosis − 3.0) equals 0.16 As a consequence of symmetry, the mean, median, and the mode are all equal for a normal random variable |

A multivariate distribution | the probabilities for a group of related random variables | |

There are two steps in standardizing a random variable X | Z = (X – μ)/σ | |

## Applications of the Normal Distribution

Question | Answer |
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Modern portfolio theory (MPT) | wide use of the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return |

choosing among portfolios using Roy’s criterion | Calculate each portfolio’s SFRatio. Choose the portfolio with the highest SFRatio. |

if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio) | SFRatio = [E(RP) – RL]/σP |

Two mainstays in managing financial risk | Value at Risk (VaR) and stress testing/scenario analysis |

Stress testing/scenario analysis | refers to a set of techniques for estimating losses in extremely unfavorable combinations of events or scenarios |

Value at Risk | is a money measure of the minimum value of losses expected over a specified time period (for example, a day, a quarter, or a year) at a given level of probability (often 0.05 or 0.01). |

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