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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 the p.d.f. of a random variableXis as follows:

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{8}x\;\;for\;0 \le x \le 4\\0\;\;\;\;otherwise\end{array} \right.\)

a. Find the value oftsuch that Pr(X鈮�t)=1/4.

b. Find the value oftsuch that Pr(X鈮�t)=1/2.

Suppose that the c.d.f. of a random variable X is as follows:

Find and sketch the p.d.f. of X

Suppose that a random variableXhas the binomial distribution with parametersn=15 andp=0.5. Find Pr(X <6).

Suppose that a random variableXhas the uniform distribution on the interval [鈭�2,8]. Find the p.d.f. ofXand the value of Pr(0<X <7).

Suppose that either of two instruments might be used for making a certain measurement. Instrument 1 yields a measurement whose p.d.f.\({{\bf{h}}_{\bf{1}}}\)is

\({{\bf{h}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{2x}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Instrument 2 yields a measurement whose p.d.f.\({{\bf{h}}_2}\)is

\({{\bf{h}}_{\bf{2}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{3}}{{\bf{x}}^{\bf{2}}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Suppose that one of the two instruments is chosen randomly, and a measurement X is made with it.

  1. Determine the marginal p.d.f. of X.
  2. If the measurement value is\({\bf{X = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{4}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{4}}\)}}\), what is the probability that instrument 1 was used?
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