Mixing
Using joint density to calculate probability that one person dies before the other
$begingroup$
Let $T_x, T_y$ be two random variables that describe the remaining lifetime of two persons aged $x$ and $y$, respectively. The joint density of $T_x$ and $T_y$ is given by
$$f_{T_x,T_y}(s,t)=begin{cases}frac{2}{45}cdot20^{-4}cdotBig(9 cdot 20^2-(3s-t)^2Big)&,s in [0,20], tin[0,60]\0&, mathrm{otherwise}end{cases}$$
How can I use this to determine the probability that the person aged $y$ dies before the person aged $x$?
probability
asked Jan 26 at 16:26
user636610user636610
82
$endgroup$
add a comment |
$begingroup$
Let $T_x, T_y$ be two random variables that describe the remaining lifetime of two persons aged $x$ and $y$, respectively. The joint density of $T_x$ and $T_y$ is given by
$$f_{T_x,T_y}(s,t)=begin{cases}frac{2}{45}cdot20^{-4}cdotBig(9 cdot 20^2-(3s-t)^2Big)&,s in [0,20], tin[0,60]\0&, mathrm{otherwise}end{cases}$$
How can I use this to determine the probability that the person aged $y$ dies before the person aged $x$?
probability
asked Jan 26 at 16:26
user636610user636610
82
$endgroup$
add a comment |
$begingroup$
Let $T_x, T_y$ be two random variables that describe the remaining lifetime of two persons aged $x$ and $y$, respectively. The joint density of $T_x$ and $T_y$ is given by
$$f_{T_x,T_y}(s,t)=begin{cases}frac{2}{45}cdot20^{-4}cdotBig(9 cdot 20^2-(3s-t)^2Big)&,s in [0,20], tin[0,60]\0&, mathrm{otherwise}end{cases}$$
How can I use this to determine the probability that the person aged $y$ dies before the person aged $x$?
probability
asked Jan 26 at 16:26
user636610user636610
82
$endgroup$
Let $T_x, T_y$ be two random variables that describe the remaining lifetime of two persons aged $x$ and $y$, respectively. The joint density of $T_x$ and $T_y$ is given by
$$f_{T_x,T_y}(s,t)=begin{cases}frac{2}{45}cdot20^{-4}cdotBig(9 cdot 20^2-(3s-t)^2Big)&,s in [0,20], tin[0,60]\0&, mathrm{otherwise}end{cases}$$
How can I use this to determine the probability that the person aged $y$ dies before the person aged $x$?
probability
probability
asked Jan 26 at 16:26
user636610user636610
82
asked Jan 26 at 16:26
user636610user636610
82
asked Jan 26 at 16:26
user636610user636610
82
asked Jan 26 at 16:26
user636610user636610
82
asked Jan 26 at 16:26
user636610user636610
82
82
add a comment |
add a comment |
2 Answers
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$begingroup$
You need to integrate the joint density over the region where $T_y$ is less than $T_x$. I suggest drawing a picture. Your integration region is the rectangle $(s,t) in [0,20]times[0,60]$ where $s$ corresponds to $T_x$ and $t$ corresponds to $T_y$. If you draw the line $s = t$ you should be able to work out the limits of integration from there.
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
$endgroup$
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
add a comment |
$begingroup$
Draw a rectangle boundary for the joint PDF, where x axis in $[0,20]$ and y axis is $[0,60]$. We need $P(T_y<T_x)$. $y$ is smaller than $x$ below $y=x$ line. So, you need to integrate the joint in the region bounded by the lines $y=x, y=0, x=20$ as follows:
$$P(T_y<T_x)=int_{0}^{20}int_{0}^{x}f(x,y)dydx$$
answered Jan 26 at 18:52
gunesgunes
3747
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
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active
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votes
$begingroup$
You need to integrate the joint density over the region where $T_y$ is less than $T_x$. I suggest drawing a picture. Your integration region is the rectangle $(s,t) in [0,20]times[0,60]$ where $s$ corresponds to $T_x$ and $t$ corresponds to $T_y$. If you draw the line $s = t$ you should be able to work out the limits of integration from there.
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
$endgroup$
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
add a comment |
$begingroup$
You need to integrate the joint density over the region where $T_y$ is less than $T_x$. I suggest drawing a picture. Your integration region is the rectangle $(s,t) in [0,20]times[0,60]$ where $s$ corresponds to $T_x$ and $t$ corresponds to $T_y$. If you draw the line $s = t$ you should be able to work out the limits of integration from there.
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
$endgroup$
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
add a comment |
$begingroup$
You need to integrate the joint density over the region where $T_y$ is less than $T_x$. I suggest drawing a picture. Your integration region is the rectangle $(s,t) in [0,20]times[0,60]$ where $s$ corresponds to $T_x$ and $t$ corresponds to $T_y$. If you draw the line $s = t$ you should be able to work out the limits of integration from there.
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
$endgroup$
You need to integrate the joint density over the region where $T_y$ is less than $T_x$. I suggest drawing a picture. Your integration region is the rectangle $(s,t) in [0,20]times[0,60]$ where $s$ corresponds to $T_x$ and $t$ corresponds to $T_y$. If you draw the line $s = t$ you should be able to work out the limits of integration from there.
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
edited Jan 26 at 18:24
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
answered Jan 26 at 16:59
inhuretnakhtinhuretnakht
38617
38617
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
add a comment |
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
So what are my limits of integration?
$endgroup$
– user636610
Jan 26 at 17:30
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
$begingroup$
I added some additional hints above. Short answer: draw a picture and it should be clear.
$endgroup$
– inhuretnakht
Jan 26 at 18:24
add a comment |
$begingroup$
Draw a rectangle boundary for the joint PDF, where x axis in $[0,20]$ and y axis is $[0,60]$. We need $P(T_y<T_x)$. $y$ is smaller than $x$ below $y=x$ line. So, you need to integrate the joint in the region bounded by the lines $y=x, y=0, x=20$ as follows:
$$P(T_y<T_x)=int_{0}^{20}int_{0}^{x}f(x,y)dydx$$
answered Jan 26 at 18:52
gunesgunes
3747
$endgroup$
add a comment |
$begingroup$
Draw a rectangle boundary for the joint PDF, where x axis in $[0,20]$ and y axis is $[0,60]$. We need $P(T_y<T_x)$. $y$ is smaller than $x$ below $y=x$ line. So, you need to integrate the joint in the region bounded by the lines $y=x, y=0, x=20$ as follows:
$$P(T_y<T_x)=int_{0}^{20}int_{0}^{x}f(x,y)dydx$$
answered Jan 26 at 18:52
gunesgunes
3747
$endgroup$
add a comment |
$begingroup$
Draw a rectangle boundary for the joint PDF, where x axis in $[0,20]$ and y axis is $[0,60]$. We need $P(T_y<T_x)$. $y$ is smaller than $x$ below $y=x$ line. So, you need to integrate the joint in the region bounded by the lines $y=x, y=0, x=20$ as follows:
$$P(T_y<T_x)=int_{0}^{20}int_{0}^{x}f(x,y)dydx$$
answered Jan 26 at 18:52
gunesgunes
3747
$endgroup$
Draw a rectangle boundary for the joint PDF, where x axis in $[0,20]$ and y axis is $[0,60]$. We need $P(T_y<T_x)$. $y$ is smaller than $x$ below $y=x$ line. So, you need to integrate the joint in the region bounded by the lines $y=x, y=0, x=20$ as follows:
$$P(T_y<T_x)=int_{0}^{20}int_{0}^{x}f(x,y)dydx$$
answered Jan 26 at 18:52
gunesgunes
3747
answered Jan 26 at 18:52
gunesgunes
3747
answered Jan 26 at 18:52
gunesgunes
3747
answered Jan 26 at 18:52
gunesgunes
3747
3747
add a comment |
add a comment |
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