Q 5.41 Find the distribution ofR=Asin胃... [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 5.41 (page 214)

Find the distribution of $R = A \sin (胃)$ , where $A$ is a fixed constant and $胃$ is uniformly distributed on $(\frac{- 蟺}{2}, \frac{蟺}{2})$ . Such a random variable $R$ arises in the theory of ballistics. If a projectile is fired from the origin at an angle $伪$ from the earth with a speed $谓$ , then the point $R$ at which it returns to the earth can be expressed as $R = (\frac{v^{2}}{g}) \sin (2 伪)$ , where $g$ is the gravitational constant, equal to $980$ centimeters per second squared.

Short Answer

Expert verified

Therefore, it returns to the earth can be expressed as $R = (\frac{v^{2}}{g}) \sin (2 伪)$

Step by step solution

Given information:

$胃$ is uniformly distributed on $(\frac{- 蟺}{2}, \frac{蟺}{2})$

Explanation:

$f_{胃} (胃) = \frac{1}{蟺} (- \frac{蟺}{2} 鈮� 胃鈮� \frac{蟺}{2})$

$f_{胃} (胃) = 0 o t h e r w i s e$

$R = A \sin (胃)$

Where $A$ is constant

consider the C.D.F of $R$

role="math" localid="1646673144927" $F_{R} (r) = P (R 鈮� r) = P (A \sin (胃) 鈮� r) = P (胃鈮� \sin^{- 1} (\frac{r}{A})) = {鈭�}_{\frac{- 蟺}{2}}^{\sin^{- 1} (\frac{r}{A})} (\frac{1}{蟺}) d 胃 = (\frac{1}{蟺}) 脳胃_{\frac{- 蟺}{2}}^{\sin^{- 1} (\frac{r}{A})} = \frac{1}{蟺} 脳 (\sin^{- 1} (\frac{r}{A}) + (\frac{蟺}{2})) f_{R} (r) = \frac{d}{d r} F_{R} (r) = \frac{1}{蟺} 脳 \frac{d}{d r} (\sin^{- 1} (\frac{r}{A})) + 0 = \frac{1}{蟺} 脳 (\frac{1}{\sqrt{1 - {(\frac{r}{A})}^{2}}} 脳 \frac{1}{A}) f_{R} (r) = \frac{1}{蟺} 脳 (\frac{1}{\sqrt{A^{2} - r^{2}}}) - A 鈮� r 鈮� A$

Explanation:

As we know that

$V = v + a t$

Where,

$V =$ Final velocity $(0)$

$v =$ initial velocity

$0 = v \sin (伪) - g t t = \frac{v \sin (伪)}{g}$

As Ascend $-$ Descend

Total

$T_{_{0}} = \frac{2 v \sin (伪)}{g}$

Range $=$ Horizontal velocity $脳$ time of flight

Range $= v 脳 T_{0}$

$= \frac{(v \cos (伪脳 2 v \sin (伪)))}{g} = \frac{(2 v^{2} \sin (伪) \cos (伪))}{g} R = \frac{(v^{2} \sin (2 伪))}{g}$

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