Learning Materials
Features
Discover
Chapter 4: Q 15E (page 255)
Suppose thatX1, . . . ,Xnare random variables suchthat thevariance of each variable is 1 and the correlationbetween each pair of different variables is 1/4. DetermineVar(X1+. . .+Xn).
\(Var\left( {{X_1} + {X_2} + ... + {X_n}} \right) = n + \frac{{n\left( {n - 1} \right)}}{4}\)
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Suppose that one word is to be selected at random from the sentence the girl put on her beautiful red hat. If X denotes the number of letters in the word that is selected, what is the value of E(X)?
Suppose that a point\({{\bf{X}}_{\bf{1}}}\)is chosen from the uniform distribution on the interval\(\left( {{\bf{0,1}}} \right)\)and that after the value\({{\bf{X}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{1}}}\)is observed, a point\({{\bf{X}}_{\bf{2}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{1}}}{\bf{,1}}} \right)\). Suppose further that additional variables\({{\bf{X}}_{\bf{3}}}{\bf{,}}{{\bf{X}}_{\bf{4}}}{\bf{,}}...\)are generated in the same way. Generally,\({\bf{j = 1,2,}}...{\bf{,}}\)after the value\({{\bf{X}}_{\bf{j}}}{\bf{ = }}{{\bf{x}}_{\bf{j}}}\)has been observed,\({{\bf{X}}_{{\bf{j + 1}}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{j}}}{\bf{,1}}} \right)\). Find the value of\({\bf{E}}\left( {{{\bf{X}}_{\bf{n}}}} \right)\).
LetYbe a discrete random variable whose p.f. is the
functionfin Example 4.1.4. LetX= |Y|. Prove that the
distribution ofXhas the p.d.f. in Example 4.1.5
\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{2}}\left| {\bf{x}} \right|\left( {\left| {\bf{x}} \right|{\bf{ + 1}}} \right)}}{\bf{,x = \pm 1, \pm 2 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)
\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{x}}\left( {{\bf{x + 1}}} \right)}}{\bf{,x = 1,2,3 \ldots ,}}\\{\bf{0,Otherwise}}\end{array} \right.\)
Suppose that an observed value of X is equally likely to come from a continuous distribution for which the pdf is for from one for which the pdf is g. Suppose that \(f\left( x \right) > 0\) for \(0 < x < 1\) and \(f\left( x \right) = 0\) otherwise, and suppose also that \(g\left( x \right) > 0\) for \(2 < x < 4\) and \(g\left( x \right) = 0\) otherwise. Determine:
Let X be a random variable. Suppose that there exists a number m such that \(\Pr \left( {X < m} \right) < \frac{1}{2}\)and\(\Pr \left( {X > m} \right) < \frac{1}{2}\).Prove that m is the unique median of the distribution of X.
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine