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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 X1 x2 x3
∏ (a) x y=sin x X1 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, =crosssectional 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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 Spring '13
 Dr.RajibMaity
 Civil Engineering

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