Unformatted text preview: (x ) = √ 2π
x −∞ e−t 2 /2 dt is important in statistics. Prove that the integral on the right converges for all real x. Exercises 59–62: Laplace transforms. Let f be continuous on [0, ∞). The Laplace transform of f is the function F deﬁned by F (s) =
0 ∞ e−sx f (x) dx. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F . 61. f (x) = 1. 63. f (x) = cos 2x. 62. f (x) = x. 64. f (x) = eax . Exercises 63–66: Probability density functions. A nonnegative function f deﬁned on ( − ∞, ∞) is called a probability density function if
∞ −∞ f (x) dx = 1. 65. Show that the function f deﬁned by f (x ) = 6x/(1 + 3x2 )2 0 x≥0 x<0 51.
1 ∞ dx. √ 1 + x5 (1 + x...
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