Prove that random variable has standard normal distribution [closed]
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How do I prove that random variable X has a standard normal distribution given the probability density function?
A random variable X has a standard normal distribution if X is absolutely continuous with density given by:
$$frac{dmathbb{P}_X}{dlambda_1}(x)=frac{1}{sqrt{2pi}}e^{-frac{1}{2} x^2},: xinmathbb{R}.$$
Provide example of probability space $(Omega, mathcal{F}, mathbb{P})$ and a random variable $X:Omegatomathbb{R}$ on $(Omega, mathcal{F}, mathbb{P})$ and verify that $X$ has a standard normal distribution.
probability-distributions random-variables
asked Nov 18 at 11:30
Thomas
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closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, Cesareo Nov 21 at 2:16
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Davide Giraudo, KReiser, user10354138, Cesareo
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up vote
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down vote
favorite
How do I prove that random variable X has a standard normal distribution given the probability density function?
A random variable X has a standard normal distribution if X is absolutely continuous with density given by:
$$frac{dmathbb{P}_X}{dlambda_1}(x)=frac{1}{sqrt{2pi}}e^{-frac{1}{2} x^2},: xinmathbb{R}.$$
Provide example of probability space $(Omega, mathcal{F}, mathbb{P})$ and a random variable $X:Omegatomathbb{R}$ on $(Omega, mathcal{F}, mathbb{P})$ and verify that $X$ has a standard normal distribution.
probability-distributions random-variables
asked Nov 18 at 11:30
Thomas
32
closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, Cesareo Nov 21 at 2:16
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Davide Giraudo, KReiser, user10354138, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How do I prove that random variable X has a standard normal distribution given the probability density function?
A random variable X has a standard normal distribution if X is absolutely continuous with density given by:
$$frac{dmathbb{P}_X}{dlambda_1}(x)=frac{1}{sqrt{2pi}}e^{-frac{1}{2} x^2},: xinmathbb{R}.$$
Provide example of probability space $(Omega, mathcal{F}, mathbb{P})$ and a random variable $X:Omegatomathbb{R}$ on $(Omega, mathcal{F}, mathbb{P})$ and verify that $X$ has a standard normal distribution.
probability-distributions random-variables
asked Nov 18 at 11:30
Thomas
32
How do I prove that random variable X has a standard normal distribution given the probability density function?
A random variable X has a standard normal distribution if X is absolutely continuous with density given by:
$$frac{dmathbb{P}_X}{dlambda_1}(x)=frac{1}{sqrt{2pi}}e^{-frac{1}{2} x^2},: xinmathbb{R}.$$
Provide example of probability space $(Omega, mathcal{F}, mathbb{P})$ and a random variable $X:Omegatomathbb{R}$ on $(Omega, mathcal{F}, mathbb{P})$ and verify that $X$ has a standard normal distribution.
probability-distributions random-variables
probability-distributions random-variables
asked Nov 18 at 11:30
Thomas
32
asked Nov 18 at 11:30
Thomas
32
asked Nov 18 at 11:30
Thomas
32
asked Nov 18 at 11:30
Thomas
32
asked Nov 18 at 11:30
Thomas
32
32
closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, Cesareo Nov 21 at 2:16
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Davide Giraudo, KReiser, user10354138, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Did, Davide Giraudo, KReiser, user10354138, Cesareo Nov 21 at 2:16
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Davide Giraudo, KReiser, user10354138, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
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1 Answer
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Let $Phi$ denote the standard normal distribution. Let $Omega =(0,1),mathcal F =$ Borel sigma algebra and let $P$ be the Lebesgue measure. Let $X(omega)=Phi ^{-1} (omega)$. ($X$ is a random variable because it is a continuous function on $(0,1))$. We have $Pr{ Xleq t}=P{omega: Phi ^{-1} (omega) leq t}=P{omega: omega leq Phi (t)}=Phi (t)$ so $X$ has distribution $Phi$.
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Let $Phi$ denote the standard normal distribution. Let $Omega =(0,1),mathcal F =$ Borel sigma algebra and let $P$ be the Lebesgue measure. Let $X(omega)=Phi ^{-1} (omega)$. ($X$ is a random variable because it is a continuous function on $(0,1))$. We have $Pr{ Xleq t}=P{omega: Phi ^{-1} (omega) leq t}=P{omega: omega leq Phi (t)}=Phi (t)$ so $X$ has distribution $Phi$.
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
add a comment |
up vote
0
down vote
accepted
Let $Phi$ denote the standard normal distribution. Let $Omega =(0,1),mathcal F =$ Borel sigma algebra and let $P$ be the Lebesgue measure. Let $X(omega)=Phi ^{-1} (omega)$. ($X$ is a random variable because it is a continuous function on $(0,1))$. We have $Pr{ Xleq t}=P{omega: Phi ^{-1} (omega) leq t}=P{omega: omega leq Phi (t)}=Phi (t)$ so $X$ has distribution $Phi$.
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Let $Phi$ denote the standard normal distribution. Let $Omega =(0,1),mathcal F =$ Borel sigma algebra and let $P$ be the Lebesgue measure. Let $X(omega)=Phi ^{-1} (omega)$. ($X$ is a random variable because it is a continuous function on $(0,1))$. We have $Pr{ Xleq t}=P{omega: Phi ^{-1} (omega) leq t}=P{omega: omega leq Phi (t)}=Phi (t)$ so $X$ has distribution $Phi$.
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
Let $Phi$ denote the standard normal distribution. Let $Omega =(0,1),mathcal F =$ Borel sigma algebra and let $P$ be the Lebesgue measure. Let $X(omega)=Phi ^{-1} (omega)$. ($X$ is a random variable because it is a continuous function on $(0,1))$. We have $Pr{ Xleq t}=P{omega: Phi ^{-1} (omega) leq t}=P{omega: omega leq Phi (t)}=Phi (t)$ so $X$ has distribution $Phi$.
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
answered Nov 18 at 11:36
Kavi Rama Murthy
43.8k31852
43.8k31852
add a comment |
add a comment |
