Q.7.74 In Example 6c, suppose that X is... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 7: Q.7.74 (page 358)

In Example 6c, suppose that $X$ is uniformly distributed over $(0, 1)$ . If the discretized regions are determined by $a_{0} = 0, a_{1} = \frac{1}{2}$ and $a_{2} = 1$ . calculate the optimal quantizer $Y$ and compute $E [{(X - Y)}^{2}] .$

Short Answer

Expert verified

The optimal quantizer $Y$ is $\frac{1}{2}$ . And the computation of $E [{(X - Y)}^{2}]$ is $\frac{1}{12} .$

Step by step solution

Given Information

$X =$ Uniformly distributed over $(0, 1)$

$a_{0} = 0,$

$a_{1} = \frac{1}{2}$

And $a_{2} = 1$

Explanation

If $X ~ U (0, 1) 鈬� F_{X} (x) = x; 0 < x < 1$

$鈭� Y = \{\begin{cases} y_{o} 鈥呪赌呪赌呪赌� if 鈥呪赌呪赌呪赌� 0 < x 鈮� \frac{1}{2} \\ y_{1} 鈥呪赌呪赌呪赌� if 鈥呪赌呪赌呪赌� \frac{1}{2} < x 鈮� 1 \end{cases}$

$\begin{aligned} P [Y = y_{0}] = F_{X} (\frac{1}{2}) 鈭� F_{X} (0) \end{aligned}$

$\begin{array}{r} = {鈭�}_{0}^{\frac{1}{2}} 鈥� d x 鈭� {鈭�}_{0}^{0} 鈥� d x = \frac{1}{2} \end{array}$

role="math" localid="1647484876133" $P [Y = y_{1}] = {鈭�}_{0}^{1} 鈥� d x 鈭� {鈭�}_{0}^{\frac{1}{2}} 鈥� d x$

$= 1 鈭� \frac{1}{2} = \frac{1}{2}$

$鈭� Var (Y) = 0 (鈭� P [Y = y_{0}] = P [Y = y_{1}] = \frac{1}{2})$

Explanation

Now the quantity is minimized when,

$\begin{aligned} y_{0} = E [X 鈭� I = 0] \end{aligned}$

localid="1647485421201">=鈭�012鈥�x12dx

localid="1647485427859" $= {鈭�}_{0}^{\frac{1}{2}} 鈥� 2 x d x$

localid="1647485434817" $= \frac{1}{4}$

$\begin{aligned} y_{1} = E [X 鈭� I = 1] \end{aligned}$

localid="1647485071030" $= {鈭�}_{\frac{1}{2}}^{1} 鈥� \frac{x}{2} d x$

$= {鈭�}_{\frac{1}{2}}^{1} 鈥� 2 x d x = \frac{3}{4}$

Explanation

Distribution of $Y$ is

$P [Y = \frac{1}{4}] = P [Y = \frac{3}{4}] = \frac{1}{2}$

And $Var (Y) = 0$

And $Var (X) = \frac{4 - 3}{12}$

$= \frac{1}{12}$

$\begin{array}{r} 鈭� E [(X 鈭� Y)^{2}] = Var (X) 鈭� Var (Y) \end{array}$

$= \frac{1}{12} 鈭� 0$

$= \frac{1}{12}$

Final Answer

Therefore, the optimal quantizer $Y$ is $\frac{1}{2}$ . And the computation of $E [(X - Y)^{2}]$ is $\frac{1}{12} .$

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91影视

In Example 6c, suppose that $X$ is uniformly distributed over $(0, 1)$ . If the discretized regions are determined by $a_{0} = 0, a_{1} = \frac{1}{2}$ and $a_{2} = 1$ . calculate the optimal quantizer $Y$ and compute $E [{(X - Y)}^{2}] .$

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Explanation

Final Answer

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91影视

In Example 6c, suppose that X is uniformly distributed over (0,1). If the discretized regions are determined by a0=0,a1=12 and a2=1. calculate the optimal quantizer Y and computeE[(X-Y)2].

Short Answer

Step by step solution

Given Information

Explanation

Explanation

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Calculus

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Example 6c, suppose that $X$ is uniformly distributed over $(0, 1)$ . If the discretized regions are determined by $a_{0} = 0, a_{1} = \frac{1}{2}$ and $a_{2} = 1$ . calculate the optimal quantizer $Y$ and compute $E [{(X - Y)}^{2}] .$