To find the probability density function
$begingroup$
I have done this problem in two ways and I get two different answer.Which one is correct.
I provided the link to the image below.
https://drive.google.com/file/d/1acToL8QBVq05mTQ9t7TgHgedremJdydA/view?usp=drivesdk
For the probability density function
$$
f(x)=begin{cases} 20x(1-x)^3, & 0<x<1 \ 0, & text{elsewhere} end{cases}
$$
find $Pbigl(x<frac{1}{2}bigr)$.
Method A
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= int_0^{1/2} 20x(1-x^3),dx \
&= 20int_0^{1/2} (1-x)x^3,dx \
&= frac{13}{16}
end{align}
Method B
For continuous distribution $Pbigl(x<frac{1}{2}bigr)=Pbigl(xlefrac{1}{2}bigr)$ so
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= fBigl(frac{1}{2}Bigr) \
&= 20Bigl(frac{1}{2}Bigr)Bigl(1-frac{1}{2}Bigr)^3 \
&= frac{10}{8}
end{align}
probability-distributions
edited Dec 31 '18 at 14:54
egreg
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
$endgroup$
add a comment |
$begingroup$
I have done this problem in two ways and I get two different answer.Which one is correct.
I provided the link to the image below.
https://drive.google.com/file/d/1acToL8QBVq05mTQ9t7TgHgedremJdydA/view?usp=drivesdk
For the probability density function
$$
f(x)=begin{cases} 20x(1-x)^3, & 0<x<1 \ 0, & text{elsewhere} end{cases}
$$
find $Pbigl(x<frac{1}{2}bigr)$.
Method A
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= int_0^{1/2} 20x(1-x^3),dx \
&= 20int_0^{1/2} (1-x)x^3,dx \
&= frac{13}{16}
end{align}
Method B
For continuous distribution $Pbigl(x<frac{1}{2}bigr)=Pbigl(xlefrac{1}{2}bigr)$ so
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= fBigl(frac{1}{2}Bigr) \
&= 20Bigl(frac{1}{2}Bigr)Bigl(1-frac{1}{2}Bigr)^3 \
&= frac{10}{8}
end{align}
probability-distributions
edited Dec 31 '18 at 14:54
egreg
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
$endgroup$
2
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00
add a comment |
$begingroup$
I have done this problem in two ways and I get two different answer.Which one is correct.
I provided the link to the image below.
https://drive.google.com/file/d/1acToL8QBVq05mTQ9t7TgHgedremJdydA/view?usp=drivesdk
For the probability density function
$$
f(x)=begin{cases} 20x(1-x)^3, & 0<x<1 \ 0, & text{elsewhere} end{cases}
$$
find $Pbigl(x<frac{1}{2}bigr)$.
Method A
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= int_0^{1/2} 20x(1-x^3),dx \
&= 20int_0^{1/2} (1-x)x^3,dx \
&= frac{13}{16}
end{align}
Method B
For continuous distribution $Pbigl(x<frac{1}{2}bigr)=Pbigl(xlefrac{1}{2}bigr)$ so
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= fBigl(frac{1}{2}Bigr) \
&= 20Bigl(frac{1}{2}Bigr)Bigl(1-frac{1}{2}Bigr)^3 \
&= frac{10}{8}
end{align}
probability-distributions
edited Dec 31 '18 at 14:54
egreg
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
$endgroup$
I have done this problem in two ways and I get two different answer.Which one is correct.
I provided the link to the image below.
https://drive.google.com/file/d/1acToL8QBVq05mTQ9t7TgHgedremJdydA/view?usp=drivesdk
For the probability density function
$$
f(x)=begin{cases} 20x(1-x)^3, & 0<x<1 \ 0, & text{elsewhere} end{cases}
$$
find $Pbigl(x<frac{1}{2}bigr)$.
Method A
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= int_0^{1/2} 20x(1-x^3),dx \
&= 20int_0^{1/2} (1-x)x^3,dx \
&= frac{13}{16}
end{align}
Method B
For continuous distribution $Pbigl(x<frac{1}{2}bigr)=Pbigl(xlefrac{1}{2}bigr)$ so
begin{align}
PBigl(x<frac{1}{2}Bigr)
&= fBigl(frac{1}{2}Bigr) \
&= 20Bigl(frac{1}{2}Bigr)Bigl(1-frac{1}{2}Bigr)^3 \
&= frac{10}{8}
end{align}
probability-distributions
probability-distributions
edited Dec 31 '18 at 14:54
egreg
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
edited Dec 31 '18 at 14:54
egreg
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
edited Dec 31 '18 at 14:54
egreg
186k1486209
edited Dec 31 '18 at 14:54
egreg
186k1486209
edited Dec 31 '18 at 14:54
egreg
186k1486209
186k1486209
asked Dec 31 '18 at 8:44
user218102user218102
234
asked Dec 31 '18 at 8:44
user218102user218102
234
asked Dec 31 '18 at 8:44
user218102user218102
234
234
2
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00
add a comment |
2
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00
2
2
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00
add a comment |
1 Answer
1
active
oldest
votes
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First method is the right one. In the second method you have confused the density function $f$ with the cumulative distribution function $F$. $P{Xleq frac 1 2}=F(frac 1 2)$ which is not the same as $f(frac 1 2)$.
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
$endgroup$
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
add a comment |
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1 Answer
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First method is the right one. In the second method you have confused the density function $f$ with the cumulative distribution function $F$. $P{Xleq frac 1 2}=F(frac 1 2)$ which is not the same as $f(frac 1 2)$.
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
$endgroup$
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
add a comment |
$begingroup$
First method is the right one. In the second method you have confused the density function $f$ with the cumulative distribution function $F$. $P{Xleq frac 1 2}=F(frac 1 2)$ which is not the same as $f(frac 1 2)$.
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
$endgroup$
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
add a comment |
$begingroup$
First method is the right one. In the second method you have confused the density function $f$ with the cumulative distribution function $F$. $P{Xleq frac 1 2}=F(frac 1 2)$ which is not the same as $f(frac 1 2)$.
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
$endgroup$
First method is the right one. In the second method you have confused the density function $f$ with the cumulative distribution function $F$. $P{Xleq frac 1 2}=F(frac 1 2)$ which is not the same as $f(frac 1 2)$.
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
answered Dec 31 '18 at 8:47
Kavi Rama MurthyKavi Rama Murthy
75.9k53270
75.9k53270
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
add a comment |
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
Thanks for your answer.But can you tell me about its difference.It be great help.
$endgroup$
– user218102
Dec 31 '18 at 8:49
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
$begingroup$
@user218102 Consider, for example, the uniform distribution on $(0,1)$. Here $f(x)=1$ for $x leq 0$ and $0$ for other values of $x$. But $F(x)$ is $0$ for $x leq 0$, $x$ for $0<x<1$ and $1$ for $x geq 1$. In general, $f$ and $F$ are related by the equation $F(x)=int_{-infty}^{x} f(t), dt$. If $X$ is a random variable with this distribution then $P{Xleq x}=F(x)=int_{-infty}^{x} f(t), dt$.
$endgroup$
– Kavi Rama Murthy
Dec 31 '18 at 8:57
add a comment |
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2
$begingroup$
You should upload images with the provided interface, rather than linking other sites that could disappear. In this particular case it wouldn't be too difficult to add the work using MathJax.
$endgroup$
– egreg
Dec 31 '18 at 14:46
$begingroup$
@egreg they linked using google docs. It can't/won't disappear unless the OP makes it disappear. Although I would say that for new users such as in this case mathjax is not easy to use. Not even remotely.
$endgroup$
– The Great Duck
Jan 1 at 9:05
$begingroup$
Honestly I don't know how to use mathjax.It seems complicated to me.
$endgroup$
– user218102
Jan 1 at 9:41
$begingroup$
@user218102 I added it for you.
$endgroup$
– egreg
Jan 1 at 9:55
$begingroup$
Thanks sir. It means alot.
$endgroup$
– user218102
Jan 1 at 10:00