Grinstead-Snell-Probability · Problem 6.3.22

Sums of Random Variables

Mathematics · conditional-density

Problem

A point YY is chosen at random from [0,1][0, 1]. A second point XX is then chosen from the interval [0,Y][0, Y]. Find the density for XX. Hint: Calculate fX∣Yf_{X|Y} as in Exercise 21 and then use fX(x)=∫x1fX∣Y(x∣y)fY(y) dy .f_X(x) = \int_x^1 f_{X|Y}(x|y) f_Y(y)\,dy\,. Can you also derive your result geometrically?

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