071.A continuous random variable X has cdffor x < 0,x² for 0 < x < 1,aF(x)=for x > 1.(a) Determine the constants a and b.a=0;b=1a=1;b=0a=3;b=7a=2;b=5

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Answered: 07 1. A continuous random variable X…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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ans Theorem 9.3. Let X and Y be independent random variables with finite variances, and a, b ER. Then Var(aX) = a²Var (X), Var (X+Y) = VarX + Var Y, Var (ax + bY) = a²VarX + b²Var Y. sercise Prove the theorem. Remark 9.2. Independence is sufficient for the variance of the sum to be equal to the sum of the variances, but not necessary. Remark 9.3. Linearity should not hold, since variance is a quadratic quantity. Remark 9.4. Note, in particular, that Var (-X) = Var(X). This is as expected, since switching the sign should not alter the spread of the distribution.
Exercise 10. For an exponential random variable X (Definition 10), show that E[X] = }. More generally, for a function o : R R, E(¢(X)] = | (u) dF(u) (5.4)
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07 1. A continuous random variable X has cdf for x < 0, x² for 0 < x < 1, a F(x) = for x > 1. (a) Determine the constants a and b. a=0;b=1 a=1;b=0 a=3;b=7 a=2;b=5