Q.4.17 Suppose that the distribution fu... [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 4: Q.4.17 (page 164)

Suppose that the distribution function of X given by
$F (b) = \{\begin{cases} 0 鈥呪赌呪赌呪赌� b < 0 \\ \frac{b}{4} 鈥呪赌呪赌呪赌� 0 鈮� b < 1 \\ \frac{1}{2} + \frac{b 鈭� 1}{4} 鈥呪赌呪赌呪赌� 1 鈮� b < 2 \\ \frac{11}{12} 鈥呪赌呪赌呪赌� 2 鈮� b < 3 \\ 1 鈥呪赌呪赌呪赌� 3 鈮� b \end{cases}$
(a) Find $P {X = i}, i = 1, 2, 3$ .
(b) Find $P \{\frac{1}{2} < X < \frac{3}{2}\}$ .

Short Answer

Expert verified

(a) The value for the P{x=i} if i=1,2,3 are

$P {1} = \frac{1}{4}$

$P {2} = \frac{1}{6}$

$P {3} = \frac{1}{12}$

(b) The value for the $P \{\frac{1}{2} < X < \frac{3}{2}\}$ is $\frac{1}{2}$ .

Step by step solution

Given information (Part a)

The distribution function of X given as

$F (b) = \{\begin{cases} 0 鈥呪赌呪赌呪赌� b < 0 \\ \frac{b}{4} 鈥呪赌呪赌呪赌� 0 鈮� b < 1 \\ \frac{1}{2} + \frac{b 鈭� 1}{4} 鈥呪赌呪赌呪赌� 1 鈮� b < 2 \\ \frac{11}{12} 鈥呪赌呪赌呪赌� 2 鈮� b < 3 \\ 1 鈥呪赌呪赌呪赌� 3 鈮� b \end{cases}$

Solution (Part a)

The calculation is given below,

$P (X = 1) = F (1 +) 鈭� F (1 鈭�)$

$= \frac{1}{2} + \frac{1 鈭� 1}{4} 鈭� \frac{1}{4}$

$= \frac{1}{4}$

$P (X = 2) = F (2 +) 鈭� F (2 鈭�)$

$= \frac{11}{12} 鈭� \frac{1}{2} 鈭� \frac{2 鈭� 1}{4}$

$= \frac{11}{12} 鈭� \frac{1}{2} 鈭� \frac{1}{4}$

$= \frac{11 鈭� 6 鈭� 3}{12}$

$= \frac{1}{6}$

Similarly,

$P (X = 3) = 1 鈭� \frac{11}{12}$

$= \frac{1}{12}$

Final answer (Part a)

The value for the P{x=i} if i=1,2,3 are

$P {1} = \frac{1}{4}$

$P {2} = \frac{1}{6}$

$P {3} = \frac{1}{12}$

Given information (Part b)

Given in the question that

Solution (Part b)

The calculation is given below,

$P (\frac{1}{2} < X < \frac{3}{2}) = F (3 / 2) 鈭� F (1 / 2)$

$= [\frac{1}{2} + \frac{\frac{3}{2} 鈭� 1}{4}] 鈭� [\frac{\frac{1}{2}}{4}]$

$= \frac{1}{2} + (\frac{1}{2} 脳 \frac{1}{4}) 鈭� (\frac{1}{2} 脳 \frac{1}{4})$

$= \frac{1}{2}$

Final answer (Part b)

The value for the $P \{\frac{1}{2} < X < \frac{3}{2}\}$ is $\frac{1}{2}$ .

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Short Answer

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Given information (Part a)

Solution (Part a)

Final answer (Part a)

Given information (Part b)

Solution (Part b)

Final answer (Part b)

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91影视

Suppose that the distribution function of X given byF(b)=0鈥呪赌呪赌呪赌�b<0b4鈥呪赌呪赌呪赌�0鈮�b<112+b鈭�14鈥呪赌呪赌呪赌�1鈮�b<21112鈥呪赌呪赌呪赌�2鈮�b<31鈥呪赌呪赌呪赌�3鈮�b(a) Find P{X=i},i=1,2,3.(b) Find P12<X<32.

Short Answer

Step by step solution

Given information (Part a)

Solution (Part a)

Final answer (Part a)

Given information (Part b)

Solution (Part b)

Final answer (Part b)

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Recommended explanations on Math Textbooks

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