Question

Show that the function f(x)=1/(x^2+π^2 ) can be taken as a probability density (distibution) function of a random variable X. Find p(X>π). Find also the cumulative distribution function F(x) of the random variable X. Find, finally, mean and standard deviation of the random variable X

1 Show that the function f(x) = can be taken as a probability density (distibution) x²+x² function of a random variable X. Find p(x > 1). Find also the cumulative distribution function F(x) of the random variable X. Find, finally, mean and standard deviation of the random variable X.

Answer 1

a As f(x) 7,0 for all x 7,0 and I fulde = x da xta2 20 / dx x²+22 - 2 - o) - Thus fex) = valid RDA. canca x2 P(xya) = f from Pflar da dx= 7 x² + x2 dx + 1/4 = 0.25 The CDF is given by F(x) = f (x2x) Adt F(x) = f fedt 2 ta²

go T * = doesn't site. Flala tan tan Flal (tan") + 5) x dia Mean; 4= E(X) - jafx) dx = x²+72 - ll=0; is Odd function o Also E(x²) = x fande = x + 2 da x ²7² X 2 de x+22) which diverges. so standard deviation - / [(x) (E(x)) ² Doesn't exist. Hance Mean le=0 standard deviation 1