Generating function of random variable $X$
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Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
Is this easy to show by definition or how do I solve it?
probability probability-distributions generating-functions
asked Aug 20 at 9:14
mandella
713420
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Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
Is this easy to show by definition or how do I solve it?
probability probability-distributions generating-functions
asked Aug 20 at 9:14
mandella
713420
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
Is this easy to show by definition or how do I solve it?
probability probability-distributions generating-functions
asked Aug 20 at 9:14
mandella
713420
Suppose random variable $X$ with values $2,3,...$ has distribution $P(X=k) = (k-1)p^2(1-p)^k-2$. Show that $$G_x(s) = left(fracps1-s(1-p)right)^2$$
Is this easy to show by definition or how do I solve it?
probability probability-distributions generating-functions
asked Aug 20 at 9:14
mandella
713420
asked Aug 20 at 9:14
mandella
713420
asked Aug 20 at 9:14
mandella
713420
asked Aug 20 at 9:14
mandella
713420
713420
add a comment |Â
add a comment |Â
2 Answers
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Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
got it thank you
– mandella
Aug 20 at 9:29
add a comment |Â
up vote
0
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Hint Remember the definition of $G_X$ :
$$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$
which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.
answered Aug 20 at 9:23
tmaths
1,326113
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
got it thank you
– mandella
Aug 20 at 9:29
add a comment |Â
up vote
1
down vote
Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
got it thank you
– mandella
Aug 20 at 9:29
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
Fot $0<t<1$, $sum_k=0^infty t^k= frac 1 1-t$ and $sum_k=0^infty kt^k-1= -frac 1 (1-t)^2$. This gives $sum_k=0^infty kt^k= -frac t (1-t)^2$. Now write down $sum t^kPX=k$ and use these formulas to compute $G_X(t)$.
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
answered Aug 20 at 9:26
Kavi Rama Murthy
23.3k2933
23.3k2933
got it thank you
– mandella
Aug 20 at 9:29
add a comment |Â
got it thank you
– mandella
Aug 20 at 9:29
got it thank you
– mandella
Aug 20 at 9:29
got it thank you
– mandella
Aug 20 at 9:29
add a comment |Â
up vote
0
down vote
Hint Remember the definition of $G_X$ :
$$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$
which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.
answered Aug 20 at 9:23
tmaths
1,326113
add a comment |Â
up vote
0
down vote
Hint Remember the definition of $G_X$ :
$$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$
which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.
answered Aug 20 at 9:23
tmaths
1,326113
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint Remember the definition of $G_X$ :
$$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$
which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.
answered Aug 20 at 9:23
tmaths
1,326113
Hint Remember the definition of $G_X$ :
$$ G_X(s) = sum_n=0^infty mathbb P (X=k)s^k $$
which is defined on $[-1,1]$. So with that definition, and by making appear a geometric sum (or something close to it) you might be able to find the answer.
answered Aug 20 at 9:23
tmaths
1,326113
answered Aug 20 at 9:23
tmaths
1,326113
answered Aug 20 at 9:23
tmaths
1,326113
answered Aug 20 at 9:23
tmaths
1,326113
1,326113
add a comment |Â
add a comment |Â
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