Probability generating function of a discrete variate process with binomial thinning
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An $mathbfAR(1)$ is given as:
$X_t = alpha * X_t-1 + Z_t $
If $X_T$ is a discrete r.v and $Z_t $ cannot be independent of $X_t-1$, we use the scalar multiplication to denote the binomial thinning.
$X_t = alpha * X_t-1 +z_T $
I want to prove its probability density function as
$P_z(s)=E(s^z) = P_x(s)/P_x(1-alpha + alpha s)$
I stared as follows:
Specifying $z_t$:
$Z_t = X_t - alpha * X_t-1$
$E(S^z_t) = fracE(s^X_t)E(s^alpha X_t-1) = fracE(s^X_t)E(s^alpha s^X_t-1) $
The numerator seems to resemble to the proof, but what about the denominator?
probability discrete-mathematics probability-distributions
asked Aug 25 at 7:52
Tos Hina
1,027418
add a comment |Â
up vote
0
down vote
favorite
An $mathbfAR(1)$ is given as:
$X_t = alpha * X_t-1 + Z_t $
If $X_T$ is a discrete r.v and $Z_t $ cannot be independent of $X_t-1$, we use the scalar multiplication to denote the binomial thinning.
$X_t = alpha * X_t-1 +z_T $
I want to prove its probability density function as
$P_z(s)=E(s^z) = P_x(s)/P_x(1-alpha + alpha s)$
I stared as follows:
Specifying $z_t$:
$Z_t = X_t - alpha * X_t-1$
$E(S^z_t) = fracE(s^X_t)E(s^alpha X_t-1) = fracE(s^X_t)E(s^alpha s^X_t-1) $
The numerator seems to resemble to the proof, but what about the denominator?
probability discrete-mathematics probability-distributions
asked Aug 25 at 7:52
Tos Hina
1,027418
You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
An $mathbfAR(1)$ is given as:
$X_t = alpha * X_t-1 + Z_t $
If $X_T$ is a discrete r.v and $Z_t $ cannot be independent of $X_t-1$, we use the scalar multiplication to denote the binomial thinning.
$X_t = alpha * X_t-1 +z_T $
I want to prove its probability density function as
$P_z(s)=E(s^z) = P_x(s)/P_x(1-alpha + alpha s)$
I stared as follows:
Specifying $z_t$:
$Z_t = X_t - alpha * X_t-1$
$E(S^z_t) = fracE(s^X_t)E(s^alpha X_t-1) = fracE(s^X_t)E(s^alpha s^X_t-1) $
The numerator seems to resemble to the proof, but what about the denominator?
probability discrete-mathematics probability-distributions
asked Aug 25 at 7:52
Tos Hina
1,027418
An $mathbfAR(1)$ is given as:
$X_t = alpha * X_t-1 + Z_t $
If $X_T$ is a discrete r.v and $Z_t $ cannot be independent of $X_t-1$, we use the scalar multiplication to denote the binomial thinning.
$X_t = alpha * X_t-1 +z_T $
I want to prove its probability density function as
$P_z(s)=E(s^z) = P_x(s)/P_x(1-alpha + alpha s)$
I stared as follows:
Specifying $z_t$:
$Z_t = X_t - alpha * X_t-1$
$E(S^z_t) = fracE(s^X_t)E(s^alpha X_t-1) = fracE(s^X_t)E(s^alpha s^X_t-1) $
The numerator seems to resemble to the proof, but what about the denominator?
probability discrete-mathematics probability-distributions
asked Aug 25 at 7:52
Tos Hina
1,027418
asked Aug 25 at 7:52
Tos Hina
1,027418
asked Aug 25 at 7:52
Tos Hina
1,027418
asked Aug 25 at 7:52
Tos Hina
1,027418
1,027418
You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14
add a comment |Â
You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14
You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14
You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14
add a comment |Â
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You can get displayed equations by enclosing the in double instead of single dollar signs. That makes them a lot easier to read, especially when you mix fractions, exponents and subscripts.
– joriki
Aug 27 at 5:14