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A point is randomly selected with uniform probability in the $X-Y$ plane within the rectangle with corners at $(0,0), (1,0), (1,2)$ and $(0,2).$ If $p$ is the length of the position vector of the point, the expected value of $p^{2}$ is

1. $\left(\dfrac{2}{3}\right)$
2. $\quad 1$
3. $\left(\dfrac{4}{3}\right)$
4. $\left(\dfrac{5}{3}\right)$

edited by
I have one doubt in the posted solutions. We have one more ways to calculate probability of each point in 2-D plane  $\rightarrow$
For ex. P = $\frac{1}{2}$  will come via two points $\left ( \frac{1}{2}, 0 \right ) and \left (0, \frac{1}{2} \right )$ respectively. Now probability of obtaining each point is $\frac{1}{2}*1$ and $\frac{1}{2}*1$ (because X and Y both are following uniform distribution and both are independent). So total probability of getting P =  $\frac{1}{2}$ is $\left ( \frac{1}{2} + \frac{1}{2} \right )$ . But via mentioned method probability for P =  $\frac{1}{2}$ will be $\frac{1}{\sqrt{5}}$ .

Where am i doing mistake ?
@Chhotu

First of all considering uniform random variable which is continuous the probability at a point is 0.So, you cannot solve like this.
edited by

Thank you @VS ji,

But first of all my following assumption is wrong ->

For ex. P = 1/2  will come via two points (1/2,0)and(0,1/2) respectively.

P = 1/2 will come via all the points in quarter circle whose radius is 1/2. But if we will do via this approach then we will not be able to cover all points in mentioned rectangle. Above diagram depicts the scenario of our Question.

The length $p$ of our point $(x,y)$ selected randomly in $XY$ plane, from origin is given by

$p=\sqrt{x^{2} + y^{2}}$

$p^{2} = x^{2} + y^{2}$

Expected value of $p^{2}$ is given by

$E[p^{2}]=E[x^{2} + y^{2}]$

By linearity of expectation

$E[x^{2} + y^{2}]=E[x^{2}]+E[ y^{2}]$

Now we need to calculate the probability density function of $X$ and $Y.$

Since, distribution is Uniform

$X$ goes from $0$ to $1,$ so $\textsf{PDF}(x) = \frac{1}{1-0}=1$

$Y$ goes from $0$ to $2$ so $\textsf{PDF}(y) = \frac{1}{2-0}=\frac{1}{2}$

Now we evaluate

$E[X^2]= \int_{0}^{1}x^{2}.1dx = \frac{1}{3}$

$E[Y^2]=\int_{0}^{2}y^{2}\times(1/2)dy = \frac{4}{3}$

$E[P^2]=E[X^2] + E[Y^2]= \frac{5}{3}$

Correct Answer: $D$

answer = option D reshown
Plz tell once again how u calculated the value of P(p) ?

here, $f(x) = P(p)$ why P(p) is 1/root (5)???

2 doubt-

1. why we are taking P(p)dp while calculating E(p^2), why not P(p^2)dp  ??

2. why can't we first calculate expected value of P, and then take square of it. ??

It is asked to calculate expectation of p^2, i.e varience of p

Let A = (x,y) be any point in that rectangular plane. The length of the position vector p is $p = \sqrt{x^{2} + y^{2}}$

Therefore,

$E(p^{2}) = E(x^{2}+y^{2}) = E(x^{2})+E(y^{2})$              ...........(1)

Now, x and y are random variables distributed uniformly over (0,1) and (0,2) respectively.

We know, that for a random variable X distributed uniformly over (a,b),

$E(X) = \frac{a+b}{2}$ and $Var(X) = \frac{(b-a)^{2}}{12}$

Also, for any random variable, in general, $Var(X) = E(X^{2}) - E(X)^{2}$

For the random variable x,

$E(x) = \frac{0+1}{2} = \frac{1}{2}$

$Var(x) = \frac{(1-0)^{2}}{12} = \frac{1}{12}$

$\therefore E(x^{2}) = Var(x) + E(x)^{2} = \frac{1}{12}+\frac{1}{4} = \frac{1}{3}$              ...........(2).

For the random variable y,

$E(y) = \frac{0+2}{2} = 1$

$Var(y) = \frac{(2-0)^{2}}{12} = \frac{1}{3}$

$\therefore E(y^{2}) = Var(x) + E(x)^{2} = \frac{1}{3}+1 = \frac{4}{3}$              ...........(3)

Using (2) and (3) in (1), we get:-

$E(p^{2}) = E(x^{2}) + E(y^{2}) = \frac{1}{3} + \frac{4}{3} = \frac{5}{3}$

Hence, the answer is D.

Best and most simple !
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