Moment generation function - compound Poisson
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Let $S_1$ be distributed with compound Poisson $lambda_1=2$ and discrete indeminizations: $f_1 (x), x geq 0$. And let $S_2$ be compound Poisson distributed with $lambda_2=4$, and discrete indeminizations: $f_2(x)=frac{1}{2}I_{{0}}(x)+frac{1}{2}I_{]0,infty[}(x)$.
How can I prove that $S_1$ and $S_2$ have the same Moment generating function?
I know that if $S_1,S_2,..,S_m$ are independent random variables Poisson compound with parameters $lambda_i$ and let $P_i$ be the respective indeminizations. Then $S=S_1+S_2+..+S_m$ is a compound Poisson distributed with $lambda=sum_{i=1}^{m} lambda_i$ and $P(x)=sum_{i=1}^{m} frac{lambda_i}{lambda} P_i(x)$.
I also know that $m_{S_N}(t)=m_N(log(m_X(t))$ (with $S_N=sum_{i}^{N} X_i$)
Using this definitions I find that for $S_1$ , we have $m_{S_1}(t)=exp(2(M_X(t)-1))$. I don't know if this is correct, and I can't get $m_{S_2}$.
Thanks in advance!
statistics moment-generating-functions risk-assessment
asked Nov 11 at 19:29
Amateur Mathematician
191213
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up vote
1
down vote
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Let $S_1$ be distributed with compound Poisson $lambda_1=2$ and discrete indeminizations: $f_1 (x), x geq 0$. And let $S_2$ be compound Poisson distributed with $lambda_2=4$, and discrete indeminizations: $f_2(x)=frac{1}{2}I_{{0}}(x)+frac{1}{2}I_{]0,infty[}(x)$.
How can I prove that $S_1$ and $S_2$ have the same Moment generating function?
I know that if $S_1,S_2,..,S_m$ are independent random variables Poisson compound with parameters $lambda_i$ and let $P_i$ be the respective indeminizations. Then $S=S_1+S_2+..+S_m$ is a compound Poisson distributed with $lambda=sum_{i=1}^{m} lambda_i$ and $P(x)=sum_{i=1}^{m} frac{lambda_i}{lambda} P_i(x)$.
I also know that $m_{S_N}(t)=m_N(log(m_X(t))$ (with $S_N=sum_{i}^{N} X_i$)
Using this definitions I find that for $S_1$ , we have $m_{S_1}(t)=exp(2(M_X(t)-1))$. I don't know if this is correct, and I can't get $m_{S_2}$.
Thanks in advance!
statistics moment-generating-functions risk-assessment
asked Nov 11 at 19:29
Amateur Mathematician
191213
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $S_1$ be distributed with compound Poisson $lambda_1=2$ and discrete indeminizations: $f_1 (x), x geq 0$. And let $S_2$ be compound Poisson distributed with $lambda_2=4$, and discrete indeminizations: $f_2(x)=frac{1}{2}I_{{0}}(x)+frac{1}{2}I_{]0,infty[}(x)$.
How can I prove that $S_1$ and $S_2$ have the same Moment generating function?
I know that if $S_1,S_2,..,S_m$ are independent random variables Poisson compound with parameters $lambda_i$ and let $P_i$ be the respective indeminizations. Then $S=S_1+S_2+..+S_m$ is a compound Poisson distributed with $lambda=sum_{i=1}^{m} lambda_i$ and $P(x)=sum_{i=1}^{m} frac{lambda_i}{lambda} P_i(x)$.
I also know that $m_{S_N}(t)=m_N(log(m_X(t))$ (with $S_N=sum_{i}^{N} X_i$)
Using this definitions I find that for $S_1$ , we have $m_{S_1}(t)=exp(2(M_X(t)-1))$. I don't know if this is correct, and I can't get $m_{S_2}$.
Thanks in advance!
statistics moment-generating-functions risk-assessment
asked Nov 11 at 19:29
Amateur Mathematician
191213
Let $S_1$ be distributed with compound Poisson $lambda_1=2$ and discrete indeminizations: $f_1 (x), x geq 0$. And let $S_2$ be compound Poisson distributed with $lambda_2=4$, and discrete indeminizations: $f_2(x)=frac{1}{2}I_{{0}}(x)+frac{1}{2}I_{]0,infty[}(x)$.
How can I prove that $S_1$ and $S_2$ have the same Moment generating function?
I know that if $S_1,S_2,..,S_m$ are independent random variables Poisson compound with parameters $lambda_i$ and let $P_i$ be the respective indeminizations. Then $S=S_1+S_2+..+S_m$ is a compound Poisson distributed with $lambda=sum_{i=1}^{m} lambda_i$ and $P(x)=sum_{i=1}^{m} frac{lambda_i}{lambda} P_i(x)$.
I also know that $m_{S_N}(t)=m_N(log(m_X(t))$ (with $S_N=sum_{i}^{N} X_i$)
Using this definitions I find that for $S_1$ , we have $m_{S_1}(t)=exp(2(M_X(t)-1))$. I don't know if this is correct, and I can't get $m_{S_2}$.
Thanks in advance!
statistics moment-generating-functions risk-assessment
statistics moment-generating-functions risk-assessment
asked Nov 11 at 19:29
Amateur Mathematician
191213
asked Nov 11 at 19:29
Amateur Mathematician
191213
edited Nov 12 at 16:55
asked Nov 11 at 19:29
Amateur Mathematician
191213
asked Nov 11 at 19:29
Amateur Mathematician
191213
asked Nov 11 at 19:29
Amateur Mathematician
191213
191213
add a comment |
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