Basic function for CDFs and PDFs of continuous random variables
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Working through some problems in Introduction to Probability (Blitzstein).
Let F be the CDF of a continuous r.v., and f = F' be the PDF
(a) Show that g defined by g(x) = 2F(x)f(x) is also a valid PDF.
Does this have something to do with the fact that a PDF should integrate to 1?
(b) Show that h defined by $h(x) = frac{1}{2}f(-x) + frac{1}{2}f(x)$ is also a valid PDF.
Any hints on how to manipulate functions & their derivatives in ways that is useful for PDFs & CDFs? Are there some general concepts anyone knows?
probability-distributions random-variables
asked Dec 2 at 14:40
user603569
287
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0
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Working through some problems in Introduction to Probability (Blitzstein).
Let F be the CDF of a continuous r.v., and f = F' be the PDF
(a) Show that g defined by g(x) = 2F(x)f(x) is also a valid PDF.
Does this have something to do with the fact that a PDF should integrate to 1?
(b) Show that h defined by $h(x) = frac{1}{2}f(-x) + frac{1}{2}f(x)$ is also a valid PDF.
Any hints on how to manipulate functions & their derivatives in ways that is useful for PDFs & CDFs? Are there some general concepts anyone knows?
probability-distributions random-variables
asked Dec 2 at 14:40
user603569
287
Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Working through some problems in Introduction to Probability (Blitzstein).
Let F be the CDF of a continuous r.v., and f = F' be the PDF
(a) Show that g defined by g(x) = 2F(x)f(x) is also a valid PDF.
Does this have something to do with the fact that a PDF should integrate to 1?
(b) Show that h defined by $h(x) = frac{1}{2}f(-x) + frac{1}{2}f(x)$ is also a valid PDF.
Any hints on how to manipulate functions & their derivatives in ways that is useful for PDFs & CDFs? Are there some general concepts anyone knows?
probability-distributions random-variables
asked Dec 2 at 14:40
user603569
287
Working through some problems in Introduction to Probability (Blitzstein).
Let F be the CDF of a continuous r.v., and f = F' be the PDF
(a) Show that g defined by g(x) = 2F(x)f(x) is also a valid PDF.
Does this have something to do with the fact that a PDF should integrate to 1?
(b) Show that h defined by $h(x) = frac{1}{2}f(-x) + frac{1}{2}f(x)$ is also a valid PDF.
Any hints on how to manipulate functions & their derivatives in ways that is useful for PDFs & CDFs? Are there some general concepts anyone knows?
probability-distributions random-variables
probability-distributions random-variables
asked Dec 2 at 14:40
user603569
287
asked Dec 2 at 14:40
user603569
287
asked Dec 2 at 14:40
user603569
287
asked Dec 2 at 14:40
user603569
287
asked Dec 2 at 14:40
user603569
287
287
Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45
add a comment |
Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45
Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45
Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45
add a comment |
1 Answer
1
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0
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Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:mathbb Rtomathbb R$ is a PDF if and only if it is nonnegative and integrable with $int_{-infty}^{infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=frac12f(-x)+frac12f(x)$?
answered Dec 2 at 14:48
drhab
95.3k543126
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:mathbb Rtomathbb R$ is a PDF if and only if it is nonnegative and integrable with $int_{-infty}^{infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=frac12f(-x)+frac12f(x)$?
answered Dec 2 at 14:48
drhab
95.3k543126
add a comment |
up vote
0
down vote
Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:mathbb Rtomathbb R$ is a PDF if and only if it is nonnegative and integrable with $int_{-infty}^{infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=frac12f(-x)+frac12f(x)$?
answered Dec 2 at 14:48
drhab
95.3k543126
add a comment |
up vote
0
down vote
up vote
0
down vote
Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:mathbb Rtomathbb R$ is a PDF if and only if it is nonnegative and integrable with $int_{-infty}^{infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=frac12f(-x)+frac12f(x)$?
answered Dec 2 at 14:48
drhab
95.3k543126
Hints
on a)
If $F(x)$ is a CDF then so is $F(x)^2$.
What will be the looks of an eventual PDF for $F(x)^2$ if $F(x)$ has a derivative?
on b)
A function $k:mathbb Rtomathbb R$ is a PDF if and only if it is nonnegative and integrable with $int_{-infty}^{infty} k(x)dx=1$.
If $f$ indeed has these qualities, then what can be said about $h(x)=frac12f(-x)+frac12f(x)$?
answered Dec 2 at 14:48
drhab
95.3k543126
answered Dec 2 at 14:48
drhab
95.3k543126
answered Dec 2 at 14:48
drhab
95.3k543126
answered Dec 2 at 14:48
drhab
95.3k543126
95.3k543126
add a comment |
add a comment |
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Yes, just do it!
– GNUSupporter 8964民主女神 地下教會
Dec 2 at 14:45