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Chapter 3: Q15E (page 117)

Suppose thatXhas the p.d.f.


Find and sketch the c.d.f. ofX.

Short Answer

Expert verified

The c.d.f of X is

Step by step solution

01

Given information


02

Calculate c.d.f of X

03

Drawing graph of c.d.f

The c.d.f of X is given by:


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Most popular questions from this chapter

Let X be a random vector that is split into three parts,\(X = \left( {Y,Z,W} \right)\)Suppose that X has a continuous joint distribution with p.d.f.\(f\left( {y,z,w} \right)\).Let\({g_1}\left( {y,z|w} \right)\)be the conditional p.d.f. of (Y, Z) given W = w, and let\({g_2}\left( {y|w} \right)\)be the conditional p.d.f. of Y given W = w. Prove that\({g_2}\left( {y|w} \right) = \int {{g_1}\left( {y,z|w} \right)dz} \)

Suppose that the n variables\({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 1]and that the random variables \({{\bf{Y}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{Y}}_{\bf{n}}}\) are defined as in Eq. (3.9.8). Determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \le {\bf{0}}{\bf{.8}}} \right)\)

Suppose that the p.d.f. of a random variable X is as

follows:\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{2}x\,\,\,\,\,\,\,\,for\,0 < x < 2\\0\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)

Also, suppose that \(Y = X\left( {2 - X} \right)\) Determine the cdf and the pdf of Y .

Find the unique stationary distribution for the Markov chain in Exercise 2.

Suppose that an electronic system comprises four components, and let\({X_j}\)denote the time until component j fails to operate (j = 1, 2, 3, 4). Suppose that\({X_1},{X_2},{X_3}\)and\({X_4}\)are i.i.d. random variables, each of which has a continuous distribution with c.d.f.\(F\left( x \right)\)Suppose that the system will operate as long as both component 1 and at least one of the other three components operate. Determine the c.d.f. of the time until the system fails to operate.
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