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Continuous Random Variables Page 1 Mean of a Continuous Random Variable The goal of this activity is to understand the mean or expected value of a continuous random variable. 0.30 The figure on the right represents a relative frequency histogram of n observation organized in m intervals of width ∆x and defined by the function h(x). In addition, h(x) approximates the probability density function f(x) of a continuous random variable from a to b. y = f(x) 0.25 0.20 0.15 h(x4) 0.10 0.05 h(x3) h(x1) 0.00 0 a =1 x1 h(x2) x22 h(xm) x33 x44 5 6 . 7. . 8 9 xm10 =b 11 Recall that the mean of a discrete random variable X is given by µ X = ∑ [x ⋅ P(x )], where x is the value of the random variable and P(x) is the probability of observing the random variable x. m Thus, the mean of the relative frequency histogram is given by µ X = ∑ [ xi ⋅ h ( xi )∆x ] . i =1 Let n grow without bound, which subsequently decreases interval ∆x without bound. Then, the function h(x) approaches f(x) and we get µ X = lim ∑ xf ( xi )∆x ∆x →0 This results in the following definition. The Mean of a Continuous Random Variable The mean (or expected value) of a continuous random variable X with a probability density function f(x) is given by the formula µX = E( X ) = ∞ ∫ xf ( x )dx −∞ Robert A. Powers University of Northern Colorado Continuous Random Variables Page 2 1. Find the mean of uniform probability distribution from 0 to 1 given by the probability density function U(0,1), where 0, if x < 0 U (0,1) = 1, if 0 ≤ x < 1 0, if x ≥ 1 2. Let the random variable X be the lengths of time in minutes between calls to 911 in a small city that were reported in the newspaper over a two day period. Suppose that a reasonable probability model for X is given by the probability density function (p.d.f.) below. 1 − x / 20 f ( x) = e ,0≤x<∞ 20 Find the expected time between calls. 3. Let Y be a continuous random variable with p.d.f. g(y) = 2y, 0 < y < 1. Find the mean of Y. 4. Let Z be a continuous random variable with p.d.f h ( z ) = 1 π ( x 2 + 1) for z ∈ R. Find the mean of Z. Challenge Problem A triangle probability distribution function on the interval [a,b] is shown in the figure on the right. Find the mean of the triangle distribution. Robert A. Powers a b University of Northern Colorado Continuous Random Variables Page 3 Standard Deviation of a Continuous Random Variable The goal of this activity is to understand the variance and standard deviation of a continuous random variable. 0.30 The figure on the right represents a relative frequency histogram of n observation organized in m intervals of width ∆x and defined by the function h(x). In addition, h(x) approximates the probability density function f(x) of a continuous random variable from a to b. y = f(x) 0.25 0.20 0.15 h(x4) 0.10 0.05 h(x3) h(x1) 0.00 0 a =1 x1 h(x2) x22 h(xm) x33 x44 5 6 . 7. . 8 9 xm10 =b 11 Recall that the variance of a discrete random variable X is given by σ X2 = ∑ [( x − µ X ) 2 ⋅ P( x )] , where x is the value of the random variable, µX is the mean of the random variable, and P(x) is the probability of observing the random variable x. m Thus, the mean of the relative frequency histogram is given by σ X2 = ∑ [( xi − µ X ) 2 ⋅ h ( xi )∆x ] . i =1 Let n grow without bound, which subsequently decreases interval ∆x without bound. Then, the function h(x) approaches f(x) and we get σ X2 = lim ∑ ( xi − µ X ) 2 f ( xi ) ∆x ∆x → 0 This results in the following definition. The Variance and Standard Deviation of a Continuous Random Variable The variance of a continuous random variable X with a probability density function f(x) is given by the formula ∞ σ = Var ( X ) = E [( X − µ X ) ] = ∫ ( x − µ X ) 2 f ( x )dx 2 X 2 −∞ The standard deviation of X is σ = Var ( X ) Robert A. Powers University of Northern Colorado Continuous Random Variables Page 4 Note: an equivalent form of the variance of X is given by σ = 2 X ∞ ∫x 2 f ( x )dx − µ X2 −∞ 1. Find the standard deviation of uniform probability distribution from 0 to 1 given by the probability density function U(0,1), where 0, if x < 0 U (0,1) = 1, if 0 ≤ x < 1 0, if x ≥ 1 2. Let the random variable X be the lengths of time in minutes between calls to 911 in a small city that were reported in the newspaper over a two day period. Suppose that a reasonable probability model for X is given by the probability density function (p.d.f.) below. 1 − x / 20 f ( x) = e ,0≤x<∞ 20 Find the standard deviation of the time intervals between calls. 3. Let Y be a continuous random variable with p.d.f. g(y) = 2y, 0 < y < 1. Find the standard deviation of Y. 4. Let Z be a continuous random variable with p.d.f h ( z ) = 1 π ( x 2 + 1) for z ∈ R. Find the standard deviation of Z. Challenge Problem Find the standard deviation of the triangle distribution on the interval [a,b]. Robert A. Powers University of Northern Colorado