ample 6 symmetrical PDFs 7.3 ean and variance 545 Let f (x) be a PDF that is sym

Question

ample 6 symmetrical PDFs 7.3 ean and variance 545 Let f (x) be a PDF that is symmetric around ma. In other words, f(x) is a PDF and f x) f(a x) for all x as illustrated in Figure 216. If the mean associ ated with f(x) well defined, we expect it to equal a, the PDF should balance at this point. To verify this assertion J ooxf(x) dx the mean is well defined is convergent) and do the following: a. Show that the mean is zero if a 30. a a a b. Show that g (x) f(x +a) is a PDF. Figure 7.16 A symmetric c. For a 0, use parts and b and the change of variables u a x to find the a mean. PDF f(x) around a satisfies f(x a) f(x a) for all x. Solution a. Assume a 0. Then, friss symmetric around o: namely, fo (x) for all x For any b 0, xf (x) dx f(x) dx f(x) dx f(u) du f (x) dx change of variables u uf (u) du l xf(x) dx integral properties Since we have assumed that Jooexfo) dx is convergent, the mean equals

Explanation / Answer

here as on a=0; f(x) =f(-x)

hence it is symmetric on y axis as it is even function.

that means that area on the right side is equal to area on left side.

hence mean =0 as cause both the area equal, mean will lie on center,

Revert for further clarification if any.

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