Riemann–Stieltjes integral Problem! Help!
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Please help solve the following problem:
Find E[X] if X has the following c.d.f.
$F_X(a)= left{ begin{array}{ll}
0 & aleq 0 \
a^2 & 0leq a <1/2 \
1/4 & 1/2 leq a <1 \
1 & ageq 1 \
end{array}
right.$
Since this cdf is a mixture of a continuous and discrete system, we cannot use a simple E[x] function like $E[X]=int_{mathbb{R}}x*f(x)dx$ or $E[X]=sum_{i=0}^{n}x*p(x)$.
I know that we must use the Riemann–Stieltjes integral $E[X]=int_{mathbb{R}}x*dF(x)$. I am confused on computationally taking this integral.
I think the answer is the $E[X]= int_{0}^{1/2} x*2x dx + 1*P(X=1)=1/12+3/4$.
The $P(X=1)$ portion is because of the discontinuity when you approach x=1 from the left side the answer is 1/4 and when you approach x=1 from the right side the answer is 1. So P(X=1)=1-1/4=3/4.
Please explain if this is correct and if so how to deal with the $a in [1/2,1)$ portion. To me, it is still "continuous" and I would think it should be dealt in an integral. But the pdf is 0 so the integral is 0 which doesn't make sense.
probability statistics
asked Dec 2 at 20:36
REW
113
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REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
up vote
-1
down vote
favorite
Please help solve the following problem:
Find E[X] if X has the following c.d.f.
$F_X(a)= left{ begin{array}{ll}
0 & aleq 0 \
a^2 & 0leq a <1/2 \
1/4 & 1/2 leq a <1 \
1 & ageq 1 \
end{array}
right.$
Since this cdf is a mixture of a continuous and discrete system, we cannot use a simple E[x] function like $E[X]=int_{mathbb{R}}x*f(x)dx$ or $E[X]=sum_{i=0}^{n}x*p(x)$.
I know that we must use the Riemann–Stieltjes integral $E[X]=int_{mathbb{R}}x*dF(x)$. I am confused on computationally taking this integral.
I think the answer is the $E[X]= int_{0}^{1/2} x*2x dx + 1*P(X=1)=1/12+3/4$.
The $P(X=1)$ portion is because of the discontinuity when you approach x=1 from the left side the answer is 1/4 and when you approach x=1 from the right side the answer is 1. So P(X=1)=1-1/4=3/4.
Please explain if this is correct and if so how to deal with the $a in [1/2,1)$ portion. To me, it is still "continuous" and I would think it should be dealt in an integral. But the pdf is 0 so the integral is 0 which doesn't make sense.
probability statistics
asked Dec 2 at 20:36
REW
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Please help solve the following problem:
Find E[X] if X has the following c.d.f.
$F_X(a)= left{ begin{array}{ll}
0 & aleq 0 \
a^2 & 0leq a <1/2 \
1/4 & 1/2 leq a <1 \
1 & ageq 1 \
end{array}
right.$
Since this cdf is a mixture of a continuous and discrete system, we cannot use a simple E[x] function like $E[X]=int_{mathbb{R}}x*f(x)dx$ or $E[X]=sum_{i=0}^{n}x*p(x)$.
I know that we must use the Riemann–Stieltjes integral $E[X]=int_{mathbb{R}}x*dF(x)$. I am confused on computationally taking this integral.
I think the answer is the $E[X]= int_{0}^{1/2} x*2x dx + 1*P(X=1)=1/12+3/4$.
The $P(X=1)$ portion is because of the discontinuity when you approach x=1 from the left side the answer is 1/4 and when you approach x=1 from the right side the answer is 1. So P(X=1)=1-1/4=3/4.
Please explain if this is correct and if so how to deal with the $a in [1/2,1)$ portion. To me, it is still "continuous" and I would think it should be dealt in an integral. But the pdf is 0 so the integral is 0 which doesn't make sense.
probability statistics
asked Dec 2 at 20:36
REW
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Please help solve the following problem:
Find E[X] if X has the following c.d.f.
$F_X(a)= left{ begin{array}{ll}
0 & aleq 0 \
a^2 & 0leq a <1/2 \
1/4 & 1/2 leq a <1 \
1 & ageq 1 \
end{array}
right.$
Since this cdf is a mixture of a continuous and discrete system, we cannot use a simple E[x] function like $E[X]=int_{mathbb{R}}x*f(x)dx$ or $E[X]=sum_{i=0}^{n}x*p(x)$.
I know that we must use the Riemann–Stieltjes integral $E[X]=int_{mathbb{R}}x*dF(x)$. I am confused on computationally taking this integral.
I think the answer is the $E[X]= int_{0}^{1/2} x*2x dx + 1*P(X=1)=1/12+3/4$.
The $P(X=1)$ portion is because of the discontinuity when you approach x=1 from the left side the answer is 1/4 and when you approach x=1 from the right side the answer is 1. So P(X=1)=1-1/4=3/4.
Please explain if this is correct and if so how to deal with the $a in [1/2,1)$ portion. To me, it is still "continuous" and I would think it should be dealt in an integral. But the pdf is 0 so the integral is 0 which doesn't make sense.
probability statistics
probability statistics
asked Dec 2 at 20:36
REW
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 2 at 20:36
REW
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 3 at 0:05
asked Dec 2 at 20:36
REW
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 2 at 20:36
REW
113
asked Dec 2 at 20:36
REW
113
113
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
REW is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49
add a comment |
2
Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49
2
2
Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49
Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49
add a comment |
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REW is a new contributor. Be nice, and check out our Code of Conduct.
REW is a new contributor. Be nice, and check out our Code of Conduct.
REW is a new contributor. Be nice, and check out our Code of Conduct.
REW is a new contributor. Be nice, and check out our Code of Conduct.
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Welcome to MSE. In order to make this a good question, please edit it to include your own thoughts and context for the question. Do you know what expectation is? Do you know what a CDF is?
– T. Bongers
Dec 2 at 20:49