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PSTAT_120A_Final_Review_Sheet

PSTAT_120A_Final_Review_Sheet - PSTAT 120A Spring 2010...

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PSTAT 120A Spring 2010 Final Review Sheet For the material from chapters 1-4, see the midterm review sheet. You will not be tested on discrete distributions other than the binomial and Poisson for the final. Continuous Random Variables The continuous random variables are those with a pdf f ( x ). The pdf satisfies P ( X A ) = ˆ A f ( x ) dx, A R . For the case where A = [ a, b ], then P ( X [ a, b ]) = P ( a X b ) = ˆ b a f ( x ) dx = F ( b ) - F ( a ) . A cdf F ( x ) is more useful for continuous random variables than for discrete because of the relationships F ( x ) = ˆ x -∞ f ( y ) dy dF dx = f ( x ) . The expected value of a continuous random variable X is EX = ˆ -∞ xf ( x ) dx, and more generally for g ( X ) E [ g ( X )] = ˆ -∞ g ( x ) f ( x ) dx. The continuous distributions that you are expected to identify, both by name and when they apply, and be comfortable manipulating are: Uniform ( a, b ) The uniform distribution has 2 parameters, a , and b . X U ( a, b ) means that the pdf of X is f ( x ) = ( 1 b - a for a < x < b 0 otherwise Normal ( μ, σ 2 ) The normal distribution has 2 parameters, μ and σ 2 , which are the mean and the variance, respec- tively. X N ( μ, σ 2 ) means that the pdf of X is f ( x ) = 1 σ 2 π e - ( x - μ ) 2 2 σ 2 , -∞ < x < . The normal does not have a nice formula for its cdf F ( x ), so the values are compiled in tables for Φ( z ) the cdf of the standard normal distribution ( μ = 0 , σ = 1). The standard normal distribution is symmetric about 0, so has the property Φ( z ) = 1 - Φ( - z ). If X N ( μ, σ 2 ) then Z := X - μ σ N (0 , 1), so P ( a X b ) = P ( a - μ σ X - μ σ b - μ σ ) = Φ b - μ σ - Φ a - μ σ , 1
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which allows one to obtain probabilities for X via standard normal probabilities. The normal approximation to the binomial states that for S n Bin( n, p ), if np (1 - p ) 10, then P a S n - np p np (1 - p ) b !
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