compute P(X^2+Y^2<=1) given a joint PDF
I'm given a joint PDF
$$f_{X,Y}(x,y) = begin{cases}
1/4 & -1leq x, y leq1 \
0, & else
end{cases}$$
How might i go about computing
$P(X^2+Y^2leq1)$?
What i notice is that the random variables are independent and drawn from a normal distribution. My initial idea was to simply find the double integral but i am unsure of the limits and the way to represent the random variable as a sum of the squares
probability-theory probability-distributions
edited Dec 3 '18 at 14:55
Did
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
|
show 1 more comment
I'm given a joint PDF
$$f_{X,Y}(x,y) = begin{cases}
1/4 & -1leq x, y leq1 \
0, & else
end{cases}$$
How might i go about computing
$P(X^2+Y^2leq1)$?
What i notice is that the random variables are independent and drawn from a normal distribution. My initial idea was to simply find the double integral but i am unsure of the limits and the way to represent the random variable as a sum of the squares
probability-theory probability-distributions
edited Dec 3 '18 at 14:55
Did
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
2
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50
|
show 1 more comment
I'm given a joint PDF
$$f_{X,Y}(x,y) = begin{cases}
1/4 & -1leq x, y leq1 \
0, & else
end{cases}$$
How might i go about computing
$P(X^2+Y^2leq1)$?
What i notice is that the random variables are independent and drawn from a normal distribution. My initial idea was to simply find the double integral but i am unsure of the limits and the way to represent the random variable as a sum of the squares
probability-theory probability-distributions
edited Dec 3 '18 at 14:55
Did
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
I'm given a joint PDF
$$f_{X,Y}(x,y) = begin{cases}
1/4 & -1leq x, y leq1 \
0, & else
end{cases}$$
How might i go about computing
$P(X^2+Y^2leq1)$?
What i notice is that the random variables are independent and drawn from a normal distribution. My initial idea was to simply find the double integral but i am unsure of the limits and the way to represent the random variable as a sum of the squares
probability-theory probability-distributions
probability-theory probability-distributions
edited Dec 3 '18 at 14:55
Did
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
edited Dec 3 '18 at 14:55
Did
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
edited Dec 3 '18 at 14:55
Did
246k23220454
edited Dec 3 '18 at 14:55
Did
246k23220454
edited Dec 3 '18 at 14:55
Did
246k23220454
246k23220454
asked Dec 3 '18 at 13:56
sn3jd3r
334
asked Dec 3 '18 at 13:56
sn3jd3r
334
asked Dec 3 '18 at 13:56
sn3jd3r
334
334
Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
2
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50
|
show 1 more comment
Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
2
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50
Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
2
2
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50
|
show 1 more comment
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Sorry but how is the integral on $[-1,1]times[0,1]$ related to the domain ${(x,y)mid x^2+y^2leqslant1}$ of interest?
– Did
Dec 3 '18 at 14:07
@Did it might not be, i am probably missing some fundamental knowledge of this. I edited the original post
– sn3jd3r
Dec 3 '18 at 14:18
2
Quite generally, for $Dsubset[-1,1]^2$, what is $$P((X,Y)in D) ?$$
– Did
Dec 3 '18 at 14:23
In fact, you have a uniform distribution on a square, and the target event is its inscribed circle, so this is a simple geometric probability.
– GNUSupporter 8964民主女神 地下教會
Dec 3 '18 at 15:15
@Did Generally that would be $int_{(x,y)in D}int f_{X,Y}(x,y)dydx$
– sn3jd3r
Dec 3 '18 at 15:50