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Chapter 3: Q5E (page 174)

Prove Theorem 3.8.2. (Hint: Either apply Theorem3.8.4 or first compute the cdf. separately for a > 0 and a < 0.)

Short Answer

Expert verified

Pdf of y is \(\begin{array}{l}g\left( y \right) = \left\{ \begin{array}{l}\frac{1}{{\left| a \right|}}f\left( {\frac{{y - b}}{a}} \right)\,for - \infty < y < \infty \\0\,\,otherwise\end{array} \right.\\\end{array}\)

Step by step solution

01

Given information

X be the random variable with the pdf f such that Y=aX+b

02

calculating pdf

If Y=aX+b the inverse function is \(x = \frac{{\left( {y - b} \right)}}{a}\)

Also\(\frac{{dx}}{{dy}} = \frac{1}{a}\)

Therefore,

\(g\left( y \right) = f\left[ {\frac{1}{a}\left( {y - b} \right)} \right]\left[ {\frac{{dx}}{{dy}}} \right]\)

\(g\left( y \right) = \frac{1}{{\left| a \right|}}f\left( {\frac{{y - b}}{a}} \right)\)

Hence,Pdf of y is

\(\begin{array}{l}g\left( y \right) = \left\{ \begin{array}{l}\frac{1}{{\left| a \right|}}f\left( {\frac{{y - b}}{a}} \right)\,for - \infty < y < \infty \\0\,\,otherwise\end{array} \right.\\\end{array}\)

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Most popular questions from this chapter

Suppose that the p.d.f. of X is as given in Exercise 3. Determine the p.d.f. of\(Y = 4 - {X^3}\)

Question:Prove Theorem 3.5.6.

Let X and Y have a continuous joint distribution. Suppose that

\(\;\left\{ {\left( {x,y} \right):f\left( {x,y} \right) > 0} \right\}\)is a rectangular region R (possibly unbounded) with sides (if any) parallel to the coordinate axes. Then X and Y are independent if and only if Eq. (3.5.7) holds for all\(\left( {x,y} \right) \in R\)

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:In Example 3.4.5, compute the probability that water demandXis greater than electric demandY.

Let Xbe a random variable for which the p.d.f. is as in Exercise 5. After the value ofXhas been observed, letYbe the integer closest toX. Find the p.f. of the random variableY.
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