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Chapter 3: Q1E (page 106)

Let Xbe a random variable with the p.d.f. specified in Example 3.2.6. Compute Pr(X鈮�8/27).

Short Answer

Expert verified

The probability that X is less than or equal to 8/27 is 0.44.

Step by step solution

01

Given information

Referring to Example 3.2.6, the probability density function of a random variable X is given as,

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{2}{3}{x^{ - \frac{1}{3}}}\;\;\;for\;0 < x < 1,\\0\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

02

Compute the probability

The probability is obtained as an integral over the given interval.

The required probability is computed as,

\(\begin{aligned}{}\Pr \left( {X \le \frac{8}{{27}}} \right) &= \int\limits_0^{\frac{8}{{27}}} {\frac{2}{3}{x^{ - \frac{1}{3}}}dx} \\ &= \frac{2}{3}\left( {\frac{3}{2}{x^{\frac{2}{3}}}} \right)_0^{\frac{8}{{27}}}\\ &= {\left( {\frac{8}{{27}}} \right)^{\frac{2}{3}}}\\ &= 0.44\end{aligned}\)

Therefore, the required probability is 0.44.

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Most popular questions from this chapter

Suppose that \({{\bf{X}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}\) are i.i.d. random variables andthat each of them has a uniform distribution on theinterval [0, 1]. Find the p.d.f. of\({\bf{Y = }}{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}\).

Question:Suppose thatXandYhave a continuous joint distribution for which the joint p.d.f. is

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{k}}\;{\bf{for}}\;{\bf{a}} \le {\bf{x}} \le {\bf{b}}\;{\bf{and}}\;{\bf{c}} \le {\bf{y}} \le {\bf{d}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

wherea <b,c < d, andk >0.

Find the marginal distributions ofXandY.

Suppose that a random variable X has a uniform distribution on the interval [0, 1]. Determine the p.d.f. of (a)\({{\bf{X}}^{\bf{2}}}\), (b) \({\bf{ - }}{{\bf{X}}^{\bf{3}}}\), and (c) \({{\bf{X}}^{\frac{{\bf{1}}}{{\bf{2}}}}}\).

Question:Suppose that the joint p.d.f. ofXandYis as follows:

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}2x{e^{ - y}}\;for\;0 \le x \le 1\;and\;0 < y < \infty \\0\;otherwise\end{array} \right.\)

AreXandYindependent?

Suppose that the p.d.f. of X is as given in Exercise 3.

Determine the p.d.f. of \(Y = 3X + 2\)

See all solutions

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