The probability density function of a random variable X is f(x) = C|x|. Find C, and the value of x° such that Fx(x°) = 3/4, where F is the CDF.
The pdf is-
"f(x)=Cx, 0\\le x\\le 2"
"0 , \\text{ othervise }"
As we know that for pdfs-
"\\int_{-\\infty}^{\\infty}f(x)dx=1"
"\\int_0^2Cxdx=1"
"\\dfrac{Cx^2}{2}|_0^2=1"
"\\dfrac{4C}{2}=1\\Rightarrow C=\\dfrac{1}{2}"
Also CDF is calculating by integrating the PDf-
"F_X(x^o)=\\int_{-\\infty}^{x^o}\\dfrac{x}{2}dx"
"F_X(x^o)=\\int_{0}^{x^o}\\dfrac{x}{2}dx"
"\\dfrac{3}{4}=\\dfrac{x^2}{4}|_0^{x^o}\\Rightarrow \\dfrac{(x^o)^2}{4}=\\dfrac{3}{4}\\Rightarrow x^o=\\sqrt{3}"
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