Mixing
Derivative of CDF of Multivariate (Normal) Distribution
$begingroup$
I am trying to get derivative of some probability functions and I do not know if it would is correct to use the following method. As an example, I am interested in derivative of $f(Q)$ with respect to $Q$.
$$f(Q)=Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$$
where $epsilon_i$, $epsilon_1$, $epsilon_2$, are independent normal distribution with mean $0$ and variance $sigma^2$, and $gamma$ is independent of all $epsilon$s and is normal with mean $mu$ and standard deviation of $s^2$.
1- With abuse of notations (I know it does not make sense to have $epsilon_i$ equal to anything), can I say
$$f'(Q)=Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
-Pr(epsilon_i+gamma<Q, epsilon_2+gamma=Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
+Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma=Q),$$
where
$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$, for example, is the derivative of joint distribution of $(epsilon_i+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q) minus the
derivative of the CDF of joint distribution of $(epsilon_i+gamma,epsilon_2+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q,Q).
2- Can I simplify the first term for example as follow?
$$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)=
Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$$.
Then what is $Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$ exactly?
3- How can I simplify $f'(Q)$ to be able to numerically compute it easily?
calculus probability derivatives normal-distribution
asked Jan 9 at 13:43
Neda KhNeda Kh
567
$endgroup$
add a comment |
$begingroup$
I am trying to get derivative of some probability functions and I do not know if it would is correct to use the following method. As an example, I am interested in derivative of $f(Q)$ with respect to $Q$.
$$f(Q)=Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$$
where $epsilon_i$, $epsilon_1$, $epsilon_2$, are independent normal distribution with mean $0$ and variance $sigma^2$, and $gamma$ is independent of all $epsilon$s and is normal with mean $mu$ and standard deviation of $s^2$.
1- With abuse of notations (I know it does not make sense to have $epsilon_i$ equal to anything), can I say
$$f'(Q)=Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
-Pr(epsilon_i+gamma<Q, epsilon_2+gamma=Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
+Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma=Q),$$
where
$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$, for example, is the derivative of joint distribution of $(epsilon_i+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q) minus the
derivative of the CDF of joint distribution of $(epsilon_i+gamma,epsilon_2+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q,Q).
2- Can I simplify the first term for example as follow?
$$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)=
Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$$.
Then what is $Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$ exactly?
3- How can I simplify $f'(Q)$ to be able to numerically compute it easily?
calculus probability derivatives normal-distribution
asked Jan 9 at 13:43
Neda KhNeda Kh
567
$endgroup$
add a comment |
$begingroup$
I am trying to get derivative of some probability functions and I do not know if it would is correct to use the following method. As an example, I am interested in derivative of $f(Q)$ with respect to $Q$.
$$f(Q)=Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$$
where $epsilon_i$, $epsilon_1$, $epsilon_2$, are independent normal distribution with mean $0$ and variance $sigma^2$, and $gamma$ is independent of all $epsilon$s and is normal with mean $mu$ and standard deviation of $s^2$.
1- With abuse of notations (I know it does not make sense to have $epsilon_i$ equal to anything), can I say
$$f'(Q)=Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
-Pr(epsilon_i+gamma<Q, epsilon_2+gamma=Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
+Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma=Q),$$
where
$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$, for example, is the derivative of joint distribution of $(epsilon_i+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q) minus the
derivative of the CDF of joint distribution of $(epsilon_i+gamma,epsilon_2+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q,Q).
2- Can I simplify the first term for example as follow?
$$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)=
Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$$.
Then what is $Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$ exactly?
3- How can I simplify $f'(Q)$ to be able to numerically compute it easily?
calculus probability derivatives normal-distribution
asked Jan 9 at 13:43
Neda KhNeda Kh
567
$endgroup$
I am trying to get derivative of some probability functions and I do not know if it would is correct to use the following method. As an example, I am interested in derivative of $f(Q)$ with respect to $Q$.
$$f(Q)=Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$$
where $epsilon_i$, $epsilon_1$, $epsilon_2$, are independent normal distribution with mean $0$ and variance $sigma^2$, and $gamma$ is independent of all $epsilon$s and is normal with mean $mu$ and standard deviation of $s^2$.
1- With abuse of notations (I know it does not make sense to have $epsilon_i$ equal to anything), can I say
$$f'(Q)=Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
-Pr(epsilon_i+gamma<Q, epsilon_2+gamma=Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)
+Pr(epsilon_i+gamma<Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma=Q),$$
where
$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)$, for example, is the derivative of joint distribution of $(epsilon_i+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q) minus the
derivative of the CDF of joint distribution of $(epsilon_i+gamma,epsilon_2+gamma,frac{epsilon_i+epsilon_1}{2}+gamma)$ with respect to the first term at (Q,Q,Q).
2- Can I simplify the first term for example as follow?
$$Pr(epsilon_i+gamma=Q, epsilon_2+gamma>Q, frac{epsilon_i+epsilon_1}{2}+gamma<Q)=
Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$$.
Then what is $Pr(epsilon_i+gamma=Q, epsilon_2>epsilon_i>epsilon_1)$ exactly?
3- How can I simplify $f'(Q)$ to be able to numerically compute it easily?
calculus probability derivatives normal-distribution
calculus probability derivatives normal-distribution
asked Jan 9 at 13:43
Neda KhNeda Kh
567
asked Jan 9 at 13:43
Neda KhNeda Kh
567
edited Jan 9 at 14:17
Neda Kh
asked Jan 9 at 13:43
Neda KhNeda Kh
567
asked Jan 9 at 13:43
Neda KhNeda Kh
567
asked Jan 9 at 13:43
Neda KhNeda Kh
567
567
add a comment |
add a comment |
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