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3_Quantitative Methods

3_Quantitative Methods - Common Probability Distributions...

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Common Probability Distributions - Discrete Uniform Distribution - Binomial Distribution o Bernoulli Random Variable: A Bernoulli trial produces one of two outcomes. A binomial random variables is defined as the number of successes in n Bernoulli trials. o The binomial distribution makes the following assumptions: the probability of success is constant for all trials and the trials are independent. o A Binomial random variable is completely described by two parameters n and p. A Bernoulli random variable is a binomial random variable with n=1. o Probability of x successes in n trials is given by (1 ) x n x n p p x - - ÷ o Do Sample problems from the curriculum o Bernoulli, B(1,p) : Mean = p; Variance = p(1-p) o Binomial, B(n,p); Mean=np; Variance = np(1-p) - Continuous Random Variables o The pdf for a uniform random variable is 1 ( ) ;0 f x fora x b otherwise b a = < < - o For a continuous uniform random variable: Mean= (a+b)/2; Variance= (b- a) 2 /12 - Normal Distribution o Central Limit Theorem: Sum (and mean) of a large number of independent random variables is approximately normally distributed. o Safety First Optimal Portfolio: If returns are normally distributed the safety-first optimal portfolio maximizes the safety first ratio (similar to Sharpe Ratio). For a portfolio with a given SFRatio, the probability that its return will be less than the threshold is N(-SFRatio). - Lognormal Distribution o A random variable Y follows a lognormal distribution if its natural log is normally distributed. The two parameters of lognormal distribution are the mean and standard deviation of its associated normal distribution. o Mean = exp(μ+.5σ 2 ); Variance=exp(2μ+σ 2 )x[exp(σ 2 )-1] o If a stocks continuously compounded return is normally distributed, then future stock price is necessarily lognormally distributed.
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