This set consists of three intervals dy ydy y gx fxx

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Unformatted text preview: +dy y g(x) fX(x) 0 x1 x2 x3 dx1 dx2 dx3 Fig. 1. Various Intervals for Proof of PDF for RVs Where, , but and . From this, it follows that: The right side equals to the shaded area in the Fig.1. Since Similarly, Hence we conclude that, and thus it proves the expression for the density function. Problem 1. Given that, , , and y=sin x. Determine Solution: Probability of X falling outside the interval, . So, probability of , also f Y y 0 . Now let us refer to the figure falling outside the interval below. For any . , the equation has an infinite number of solutions where the principle solution is fX(x) x1 X-1 x2 x3 ∏ (a) x y=sin x X-1 y x1 x2 ∏ x3 x (b) Fig. 2. Function of Random Variable From the symmetry in figure (b), we can get, and so on… Also . Therefore we obtain, using the equation, . Now , We obtain, , 0<y<1. But . Except for and Thus, . fY(y) 2/∏ 1 y Fig. 3. Plot of Function f Y y Problem 2.The strain energy in an elastic bar subjected to a force is given by, where, =length of the bar, =cross-sectional area of the bar, =modulus of elasticity of the elastic material. Find the density function of . Density function of with - Solution: Rewriting the equation, Thus we get, is given by, . (An...
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