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Chapter 4: Q 12E (page 264)

Suppose that X and Y are random variables such that

\(E\left( {Y|X} \right) = aX + b\)Assuming that\(Cov\left( {X,Y} \right)\)exists and that\(0 < Var\left( X \right) < \infty \), determine expressions for a and b in terms of\(E\left( X \right)\),\(E\left( Y \right)\)and\(Cov\left( {X,Y} \right)\).

Short Answer

Expert verified

Expression for a and b in terms of \(E\left( X \right)\),\(E\left( Y \right)\)and \(Cov\left( {X,Y} \right)\) Is \(a = \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}\) and \(b = E\left( Y \right) - \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}E\left( x \right)\)

Step by step solution

01

Given information

X and Yare random variables such that \(E\left( {Y|X} \right) = aX + b\)

02

Calculate the expression for a and b 

Since

\(E\left( Y \right) = E\left( {E\left( {Y|X} \right)} \right)\)

Then

\(E\left( Y \right) = aE\left( X \right) + b\)..............(1)

Also, we know

\(E\left( {XY} \right) = aE\left( {{X^2}} \right) + bE\left( X \right)\)......(2)

Two equations solving simultaneously for an as well as b

\(a = \frac{{E\left( {XY} \right) - E\left( X \right)E\left( Y \right)}}{{E\left( {{X^2}} \right) - {{\left( {E\left( X \right)} \right)}^2}}}\)

\(a = \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}\)

Putting the values of a

\(b = E\left( Y \right) - aE\left( X \right)\)

\(b = E\left( Y \right) - \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}E\left( x \right)\)

Hence, an expression for a and b in terms of\(E\left( X \right)\),\(E\left( Y \right)\)and\(Cov\left( {X,Y} \right)\)

Is \(a = \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}\) and \(b = E\left( Y \right) - \frac{{Cov\left( {X,Y} \right)}}{{Var\left( X \right)}}E\left( x \right)\)

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Most popular questions from this chapter

Show that two random variablesXandYcannot possibly have the following properties:\(E\left( X \right) = 3\),\(E\left( Y \right) = 2\),\(E\left( {{X^2}} \right) = 10\),\(E\left( {{Y^2}} \right) = 29\), and\(E\left( {XY} \right) = 0\).

Suppose that one letter is to be selected at random from the 30 letters in the sentence given in Exercise 4. If Y denotes the number of letters in the word in which the selected letter appears, what is the value of E (Y)?

Consider a coin for which the probability of obtaining a head on each given toss is 0.3. Suppose that the coin is to be tossed 15 times, and let X denote the number of heads that will be obtained.

  1. What prediction of X has the smallest M.S.E?
  2. What prediction of X has the smallest M.A.E?

Suppose that a random variable X has a continuous distribution for which the pdf f is as follows,

\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{2x}&{{\rm{for}}\;0 < x < 1}\\0&{{\rm{otherwise}}}\end{aligned}} \right.\)

Determine the value of d that minimizes,

(a)\(E\left( {{{\left( {X - d} \right)}^2}} \right)\) and (b) \(E\left( {\left| {X - d} \right|} \right)\)

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\({\bf{E}}\left( {{{\bf{X}}_{\bf{n}}}} \right)\).
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