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SalasSV_10_07_ex

# 67 the mean of a probability density function f is

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Unformatted text preview: ) ln x dx. x2 5 −1/6 x ∞ 52. 1 ∞ 2 −x2 dx. 53. 0 ∞ dx. 54. π ∞ sin2 2x dx. x2 √ dx x + 1 ln x . is a probability density function. 66. Let k > 0. Show that the function f (x ) = ke−kx x ≥ 0 0 x < 0, 55. 1 56. e CHAPTER HIGHLIGHTS 631 is a probability density function. It is called the exponential density function. 67. The mean of a probability density function f is deﬁned as the number µ= ∞ where µ is the mean. Calculate the standard deviation for the exponential density function. 69. (Useful later) Let f be a continuous, positive, decreasing function on [1, ∞). Show that ∞ x f (x) dx. −∞ 1 f (x) dx converges iff the sequence n Calculate the mean for the exponential density function. 68. The standard deviation of a probability density function f is deﬁned as the number σ= ∞ −∞ 1/ 2 an = 1 f (x) dx (x − µ) f (x) dx 2 converges. CHAPT...
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