Cauchy distribution, transformation of univariate random variable
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I'm trying to solve a strange exercise.
"A cannon is placed on a point P=$(x_0,y_0)$ of the upper side of the Cartesian plane, i.e. $y_0>0$. The cannon shuts projectiles to the ordinate axis in random directions. We know that the angle $theta$ with which a projectile hits the x-axes is distributed uniformly in $(0,pi)$. Then we can write $thetasim Unif(0,pi)$, and we call $Q = (0,X)$ the point hit by a projectile. See figure here below for a representation:
figure: P is the position of the cannon and Q is the point hit by a projectile, $thetasim Unif(0,pi)$
Show that X, the ordinate of the Q, is a random variable with support $D_X=R$ and with density:
$f_X(x)=frac{1}{pi y_0}[frac{y_0^2}{(x-x_0)^2+y_0^2}]$
Hint 1: the slope of the line from two point $P = (x_0,y_0)$ and $Q = (x_1,y_1)$ is $m=frac{y_0-y_1}{x_0-x_1} $.
Hint 2: the function $cot(theta)$ is monotone while its inverse is not."
random-variables transformation
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
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$begingroup$
I'm trying to solve a strange exercise.
"A cannon is placed on a point P=$(x_0,y_0)$ of the upper side of the Cartesian plane, i.e. $y_0>0$. The cannon shuts projectiles to the ordinate axis in random directions. We know that the angle $theta$ with which a projectile hits the x-axes is distributed uniformly in $(0,pi)$. Then we can write $thetasim Unif(0,pi)$, and we call $Q = (0,X)$ the point hit by a projectile. See figure here below for a representation:
figure: P is the position of the cannon and Q is the point hit by a projectile, $thetasim Unif(0,pi)$
Show that X, the ordinate of the Q, is a random variable with support $D_X=R$ and with density:
$f_X(x)=frac{1}{pi y_0}[frac{y_0^2}{(x-x_0)^2+y_0^2}]$
Hint 1: the slope of the line from two point $P = (x_0,y_0)$ and $Q = (x_1,y_1)$ is $m=frac{y_0-y_1}{x_0-x_1} $.
Hint 2: the function $cot(theta)$ is monotone while its inverse is not."
random-variables transformation
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
$endgroup$
|
$begingroup$
I'm trying to solve a strange exercise.
"A cannon is placed on a point P=$(x_0,y_0)$ of the upper side of the Cartesian plane, i.e. $y_0>0$. The cannon shuts projectiles to the ordinate axis in random directions. We know that the angle $theta$ with which a projectile hits the x-axes is distributed uniformly in $(0,pi)$. Then we can write $thetasim Unif(0,pi)$, and we call $Q = (0,X)$ the point hit by a projectile. See figure here below for a representation:
figure: P is the position of the cannon and Q is the point hit by a projectile, $thetasim Unif(0,pi)$
Show that X, the ordinate of the Q, is a random variable with support $D_X=R$ and with density:
$f_X(x)=frac{1}{pi y_0}[frac{y_0^2}{(x-x_0)^2+y_0^2}]$
Hint 1: the slope of the line from two point $P = (x_0,y_0)$ and $Q = (x_1,y_1)$ is $m=frac{y_0-y_1}{x_0-x_1} $.
Hint 2: the function $cot(theta)$ is monotone while its inverse is not."
random-variables transformation
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
$endgroup$
I'm trying to solve a strange exercise.
"A cannon is placed on a point P=$(x_0,y_0)$ of the upper side of the Cartesian plane, i.e. $y_0>0$. The cannon shuts projectiles to the ordinate axis in random directions. We know that the angle $theta$ with which a projectile hits the x-axes is distributed uniformly in $(0,pi)$. Then we can write $thetasim Unif(0,pi)$, and we call $Q = (0,X)$ the point hit by a projectile. See figure here below for a representation:
figure: P is the position of the cannon and Q is the point hit by a projectile, $thetasim Unif(0,pi)$
Show that X, the ordinate of the Q, is a random variable with support $D_X=R$ and with density:
$f_X(x)=frac{1}{pi y_0}[frac{y_0^2}{(x-x_0)^2+y_0^2}]$
Hint 1: the slope of the line from two point $P = (x_0,y_0)$ and $Q = (x_1,y_1)$ is $m=frac{y_0-y_1}{x_0-x_1} $.
Hint 2: the function $cot(theta)$ is monotone while its inverse is not."
random-variables transformation
random-variables transformation
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
asked Jan 15 at 16:22
Jan m.a.Jan m.a.
11
11
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