Mixing
Show that the expected value of $X$ is equal to $E(x)$
Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$
probability summation random-variables expected-value
edited Nov 22 '18 at 16:47
greedoid
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asked Nov 22 '18 at 12:45
ManMan
162
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Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$
probability summation random-variables expected-value
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
asked Nov 22 '18 at 12:45
ManMan
162
add a comment |
Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$
probability summation random-variables expected-value
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
asked Nov 22 '18 at 12:45
ManMan
162
Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$
probability summation random-variables expected-value
probability summation random-variables expected-value
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
asked Nov 22 '18 at 12:45
ManMan
162
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
asked Nov 22 '18 at 12:45
ManMan
162
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
edited Nov 22 '18 at 16:47
greedoid
38.6k114797
38.6k114797
asked Nov 22 '18 at 12:45
ManMan
162
asked Nov 22 '18 at 12:45
ManMan
162
asked Nov 22 '18 at 12:45
ManMan
162
162
add a comment |
add a comment |
2 Answers
2
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begin{eqnarray}
E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
&=& color{red}{mathbb P{X=1}}\
&+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
&+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
& vdots &\
&+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
&=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
&=& sum_{k=1}^{15} mathbb P{Xgeq k}\
end{eqnarray}
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
add a comment |
Hint
$$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
add a comment |
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2 Answers
2
active
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votes
2 Answers
2
active
oldest
votes
active
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active
oldest
votes
begin{eqnarray}
E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
&=& color{red}{mathbb P{X=1}}\
&+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
&+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
& vdots &\
&+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
&=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
&=& sum_{k=1}^{15} mathbb P{Xgeq k}\
end{eqnarray}
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
add a comment |
begin{eqnarray}
E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
&=& color{red}{mathbb P{X=1}}\
&+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
&+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
& vdots &\
&+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
&=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
&=& sum_{k=1}^{15} mathbb P{Xgeq k}\
end{eqnarray}
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
add a comment |
begin{eqnarray}
E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
&=& color{red}{mathbb P{X=1}}\
&+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
&+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
& vdots &\
&+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
&=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
&=& sum_{k=1}^{15} mathbb P{Xgeq k}\
end{eqnarray}
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
begin{eqnarray}
E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
&=& color{red}{mathbb P{X=1}}\
&+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
&+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
& vdots &\
&+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
&=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
&=& sum_{k=1}^{15} mathbb P{Xgeq k}\
end{eqnarray}
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
answered Nov 22 '18 at 13:07
greedoidgreedoid
38.6k114797
38.6k114797
add a comment |
add a comment |
Hint
$$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
add a comment |
Hint
$$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
add a comment |
Hint
$$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
Hint
$$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
answered Nov 22 '18 at 12:49
SurbSurb
37.4k94375
37.4k94375
add a comment |
add a comment |
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