Mixing
Probability distribution problem #1
Clash Royale CLAN TAG#URR8PPP
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Let $X_1$ and $X_2$ be i.i.d random variables having $textPoi(lambda)$ distribution. Put $T=X1+2cdot X2$.
So I have to find $mathbbP(X_1+2 cdot X_2=t)$.
I can't solve this. How should I proceed.
probability probability-theory probability-distributions poisson-distribution
asked Aug 3 at 18:34
Ritam
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up vote
1
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Let $X_1$ and $X_2$ be i.i.d random variables having $textPoi(lambda)$ distribution. Put $T=X1+2cdot X2$.
So I have to find $mathbbP(X_1+2 cdot X_2=t)$.
I can't solve this. How should I proceed.
probability probability-theory probability-distributions poisson-distribution
asked Aug 3 at 18:34
Ritam
192
How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $X_1$ and $X_2$ be i.i.d random variables having $textPoi(lambda)$ distribution. Put $T=X1+2cdot X2$.
So I have to find $mathbbP(X_1+2 cdot X_2=t)$.
I can't solve this. How should I proceed.
probability probability-theory probability-distributions poisson-distribution
asked Aug 3 at 18:34
Ritam
192
Let $X_1$ and $X_2$ be i.i.d random variables having $textPoi(lambda)$ distribution. Put $T=X1+2cdot X2$.
So I have to find $mathbbP(X_1+2 cdot X_2=t)$.
I can't solve this. How should I proceed.
probability probability-theory probability-distributions poisson-distribution
asked Aug 3 at 18:34
Ritam
192
edited 2 days ago
asked Aug 3 at 18:34
Ritam
192
asked Aug 3 at 18:34
Ritam
192
asked Aug 3 at 18:34
Ritam
192
192
How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago
 |Â
show 1 more comment
How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago
How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago
 |Â
show 1 more comment
1 Answer
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oldest
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0
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The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=lfloorfract2rfloor$. Therefore $P(T=t)=e^-2lambdasum_k=0^Nfraclambda^kk!fraclambda^t-2k(t-2k)!=lambda^te^-2lambdasum_k=0^Nfraclambda^-kk!(t-2k)!$. Unfortunately this last sum doesn't seem to have a neat expression.
answered 2 days ago
herb steinberg
93529
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=lfloorfract2rfloor$. Therefore $P(T=t)=e^-2lambdasum_k=0^Nfraclambda^kk!fraclambda^t-2k(t-2k)!=lambda^te^-2lambdasum_k=0^Nfraclambda^-kk!(t-2k)!$. Unfortunately this last sum doesn't seem to have a neat expression.
answered 2 days ago
herb steinberg
93529
add a comment |Â
up vote
0
down vote
The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=lfloorfract2rfloor$. Therefore $P(T=t)=e^-2lambdasum_k=0^Nfraclambda^kk!fraclambda^t-2k(t-2k)!=lambda^te^-2lambdasum_k=0^Nfraclambda^-kk!(t-2k)!$. Unfortunately this last sum doesn't seem to have a neat expression.
answered 2 days ago
herb steinberg
93529
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=lfloorfract2rfloor$. Therefore $P(T=t)=e^-2lambdasum_k=0^Nfraclambda^kk!fraclambda^t-2k(t-2k)!=lambda^te^-2lambdasum_k=0^Nfraclambda^-kk!(t-2k)!$. Unfortunately this last sum doesn't seem to have a neat expression.
answered 2 days ago
herb steinberg
93529
The standard method to get the distribution for a sum of two independent random variables is convolution. For discrete variables this takes the form of a sum. For $X_1+2X_2=t$ an individual $(kth)$ term in the sum for $P(T=t)$ is $P(X_2=k)P(X_1=t-2k)$. The upper limit on the sum is $N=lfloorfract2rfloor$. Therefore $P(T=t)=e^-2lambdasum_k=0^Nfraclambda^kk!fraclambda^t-2k(t-2k)!=lambda^te^-2lambdasum_k=0^Nfraclambda^-kk!(t-2k)!$. Unfortunately this last sum doesn't seem to have a neat expression.
answered 2 days ago
herb steinberg
93529
edited yesterday
answered 2 days ago
herb steinberg
93529
answered 2 days ago
herb steinberg
93529
answered 2 days ago
herb steinberg
93529
93529
add a comment |Â
add a comment |Â
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How is $T$ an estimator for $lambda$ (sufficient or not)? Its mean is $3lambda$, and I don't see that it has any other properties that would make it suitable as an estimator for $lambda$.
– joriki
Aug 3 at 18:53
Please see this tutorial and reference on how to typeset math on this site.
– joriki
Aug 4 at 5:43
@joriki The question isn't about an estimator for $lambda$. It looks like a question about the distribution of a sum.
– herb steinberg
2 days ago
@herbsteinberg: That's because the OP fundamentally changed the question after I commented, without replying to my comment or marking the edit. The original question was about an estimator for $lambda$, as you can see in the edit history you get when you click on the edit time stamp above.
– joriki
2 days ago
@joriki I get it. It looks like he really wasn't sure at first what he wanted. The question in its final form is straightforward.
– herb steinberg
2 days ago