Q. 5.4 The random variable X聽has the p... [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 5: Q. 5.4 (page 176)

The random variable $X$ has the probability density function
$f (x) = \{\begin{cases} a x + b x 2 & 0 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$
If $E [X] = 6$ , find
(a) $P {X < \frac{1}{2}}$ and
(b) $V a r (X)$ .

Short Answer

Expert verified

(a) The value of $P {X < \frac{1}{2}}$ is $\frac{7}{20}$

(b) The value of $V a r (X)$ is $0.06$

Step by step solution

Find the value of P{X<12} (part a)

First and foremost, the density function must satisfy the following conditions:

$1 = {鈭�}_{鈩�} f (x) d x = {鈭�}_{0}^{1} (a x + b x^{2}) d x = a 脳 \frac{x^{2}}{2} + {b 脳 \frac{x^{3}}{3}|}_{0}^{1} = \frac{a}{2} + \frac{b}{3}$

We can deduce from the fact that $E (X) = 0.6$ ,

$0.6 = E (X) = {鈭�}_{鈩�} x f (x) d x = {鈭�}_{0}^{1} (a x^{2} + b x^{3}) d x = a 脳 \frac{x^{3}}{3} + {b 脳 \frac{x^{4}}{4}|}_{0}^{1} = \frac{a}{3} + \frac{b}{4}$

As a result, we have two unknowns and two linear equations to solve.

$\{\begin{cases} 3 a + 2 b & = 6 \\ 20 a + 15 b & = 36 \end{cases}$

$a = \frac{18}{5}$ and $b = - \frac{12}{5}$ are the results of this system.

$P (X < \frac{1}{2}) = {鈭�}_{0}^{1 / 2} \frac{18}{5} x - \frac{12}{5} x^{2} d x = \frac{9}{5} x^{2} - {\frac{4}{5} x^{3}|}_{0}^{1 / 2} = \frac{7}{20}$

Find Var(X) (part b)

Let's figure out the second moment in order to calculate the variance.

$E (X^{2}) = {鈭�}_{鈩�} x^{2} f (x) d x = {鈭�}_{0}^{1} \frac{18}{5} x^{3} - \frac{12}{5} x^{4} d x = \frac{18}{5} \frac{x^{4}}{4} - {\frac{12}{5} \frac{x^{5}}{5}|}_{0}^{1} = \frac{21}{50}$

As a result, we may say that the variance is equal to

$Var (X) = E (X^{2}) - E (X)^{2} = \frac{21}{50} - 0 . 6^{2} = 0.06$

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91影视

The random variable $X$ has the probability density function
$f (x) = \{\begin{cases} a x + b x 2 & 0 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$
If $E [X] = 6$ , find
(a) $P {X < \frac{1}{2}}$ and
(b) $V a r (X)$ .

Short Answer

Step by step solution

Find the value of P{X<12} (part a)

Find Var(X) (part b)

One App. One Place for Learning.

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91影视

The random variable Xhas the probability density functionf(x)=ax+bx20<x<10otherwiseIf E[X]=6, find(a) P{X<12}and(b) Var(X).

Short Answer

Step by step solution

Find the value of P{X<12} (part a)

Find Var(X) (part b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Decision Maths

Calculus

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The random variable $X$ has the probability density function
$f (x) = \{\begin{cases} a x + b x 2 & 0 < x < 1 \\ 0 & o t h e r w i s e \end{cases}$
If $E [X] = 6$ , find
(a) $P {X < \frac{1}{2}}$ and
(b) $V a r (X)$ .