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Chapter 8: Q5 E (page 472)

Suppose that a point(X, Y, Z)is to be chosen at random in three-dimensional space, whereX,Y, andZare independent random variables, and each has the standard normal distribution. What is the probability that the distance from the origin to the point will be less than 1 unit?

Short Answer

Expert verified

\(P\left( {{X^2} + {Y^2} + {Z^2} \le 1} \right) < 0.2\)

Step by step solution

01

Given information

X, Y, and Z are independent random variables, and each of the variables follows the standard normal distribution.

02

Calculate the probability

Since \({X^2} + {Y^2} + {Z^2}\)has \({\chi ^2}\)distributed with degrees of freedom 3.

Referring to the table at the end of the book for the answer.

Hence, \(P\left( {{X^2} + {Y^2} + {Z^2} \le 1} \right) < 0.2\).

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Most popular questions from this chapter

In the situation of Example 8.5.11, suppose that we observe\({{\bf{X}}_{\bf{1}}}{\bf{ = 4}}{\bf{.7}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}{\bf{ = 5}}{\bf{.3}}\).

  1. Find the 50% confidence interval described in Example 8.5.11.
  2. Find the interval of possible胃values consistent with the observed data.
  3. Is the 50% confidence interval larger or smaller than the set of possible胃values?
  4. Calculate the value of the random variable\({\bf{Z = }}{{\bf{Y}}_{\bf{2}}}{\bf{ - }}{{\bf{Y}}_{\bf{1}}}\)as described in Example 8.5.11.
  5. Use Eq. (8.5.15) to compute the conditional probability that\(\left| {{{{\bf{\bar X}}}_{\bf{2}}}{\bf{ - \theta }}} \right|{\bf{ < 0}}{\bf{.1}}\)givenZ isequal to the value calculated in part (d).

LetX1, . . . , Xnbe a random sample from the exponential distribution with parameter胃. Find the c.d.f. for the sampling distribution of the M.L.E. of胃. (The M.L.E. itself was found in Exercise 7 in Sec. 7.5.)

Continue the analysis in Example 8.6.2 on page 498. Compute an interval (a, b) such that the posterior probability is 0.9 that a <渭<b. Compare this interval with the 90% confidence interval from Example 8.5.4 on page 487.

Suppose that X1,鈥�.., Xn forms a random sample from the Bernoulli distribution with unknown parameter p. Show thatX虅n is an efficient estimator of p.

For the conditions of Exercise 5, how large a random sample must be taken in order that\({{\bf{E}}_{\bf{p}}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - p}}{{\bf{|}}^{\bf{2}}}} \right) \le {\bf{0}}{\bf{.01}}\) whenp=0.2?
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