times and recorded the outcome each time then the arithmetic average of those

Times and recorded the outcome each time then the

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Unformatted text preview: times and recorded the outcome each time, then the arithmetic average of those outcomes is best predicted by the expected value of the random variable. If the random variable X has n possible outcomes, x 1 ,x 2 ,...,x n , then the expected value is computed as: E [ X ] = n X i =1 x i f X ( x i ) So if X is a die roll, the expected value is: E [ X ] = 1 1 6 + 2 1 6 + 3 1 6 + 4 1 6 + 5 1 6 + 6 1 6 = 1 6 (1 + 2 + 3 + 4 + 5 + 6) = 3 . 5 4 If X is a continuous random variable, then the expected value is computed as: E [ X ] = ˆ + ∞-∞ xf X ( x ) dx Variance and Standard Deviation The variance of a random variable measures the dispersion of a random variable about its expected value. In a sense, it measures how large the variation in outcomes is expected to be. Formally, the variance is de ned as: Var ( X ) = E ( X- E [ X ]) 2 and we often simply write σ 2 X to denote the variance of the random variable X . If the random variable X has n possible outcomes, as above, then the variance is computed as: Var ( X ) = n X i =1 ( x i- E [ X ]) 2 f X ( x i ) The standard deviation is the square root of the variance, and we will often write it as σ X . As usual, when the random variable in question is obvious, we sometimes suppress the subscript notation X , and write the variance and standard deviation simply as σ 2 and σ . 3 Some Important Distributions Here, we introduce some important distributions that are often used to model random variables. Bernoulli Distribution A discrete distribution that takes a value of 1 with probability p and a value of 0 with probability 1- p . Support: { , 1 } PDF: f (0) = 1- p, f (1) = p Expected Value: E [ X ] = p 5 Variance: Var ( X ) = p (1- p ) Uniform Distribution A continuous random variable which has an equal probability of taking any value between two bounds, a and b . Support: x ∈ [ a,b ] PDF: f ( x ) = 1 b- a CDF: F ( x ) = x- a b- a Expected Value: E [ X ] = b + a 2 Variance: Var ( X ) = 1 12 ( b- a ) 2 Normal Distribution A continuous random variable whose distribution is shaped like a bell curve. Has a mean of μ and variance σ 2 . Support: x ∈ (-∞ , + ∞ ) PDF: f ( x ) = 1 σ √ 2 π exp- ( x- μ ) 2 2 σ 2 CDF: 1 2 1 + erf x- μ √ 2 σ 2 Expected Value: E [ X ] = μ Variance: Var ( X ) = σ 2 4 Covariance So far we have only dealt with single random variables. Sometimes, we may have to think about multiple random variables; and in particular, how those two random variables are related....
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