Intuitive Explanation Relationship between Random Variable, Density function and Distribution
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After slowly coming to grips with Random Variables and the distributions that they subscribe to (e.g. $X$ ~ $mathcal{U}(0,1$)) , we introduced the notion of pdf, which I believe I understand in essence, but I am rather confused when, for example, we are told that for the exponential distribution to parameter $lambda > 0$
the density $f$ can be said to be $f(x):=lambda e^{-lambda x}chi_{[0,infty[}(x)$.
I always thought that distributions were set according to a random variable. However, a random variable is not mentioned anywhere above.
My guess is that $f(x):=P(X in {x})$, where $X$ is the random variable that actually has exponential distribution to parameter $lambda$, rather than the density function $f$.
I need clarity on the terms, and perhaps an explanation on how it all fits together.
probability-theory probability-distributions
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
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|
show 1 more comment
$begingroup$
After slowly coming to grips with Random Variables and the distributions that they subscribe to (e.g. $X$ ~ $mathcal{U}(0,1$)) , we introduced the notion of pdf, which I believe I understand in essence, but I am rather confused when, for example, we are told that for the exponential distribution to parameter $lambda > 0$
the density $f$ can be said to be $f(x):=lambda e^{-lambda x}chi_{[0,infty[}(x)$.
I always thought that distributions were set according to a random variable. However, a random variable is not mentioned anywhere above.
My guess is that $f(x):=P(X in {x})$, where $X$ is the random variable that actually has exponential distribution to parameter $lambda$, rather than the density function $f$.
I need clarity on the terms, and perhaps an explanation on how it all fits together.
probability-theory probability-distributions
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
$endgroup$
1
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
1
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54
|
show 1 more comment
$begingroup$
After slowly coming to grips with Random Variables and the distributions that they subscribe to (e.g. $X$ ~ $mathcal{U}(0,1$)) , we introduced the notion of pdf, which I believe I understand in essence, but I am rather confused when, for example, we are told that for the exponential distribution to parameter $lambda > 0$
the density $f$ can be said to be $f(x):=lambda e^{-lambda x}chi_{[0,infty[}(x)$.
I always thought that distributions were set according to a random variable. However, a random variable is not mentioned anywhere above.
My guess is that $f(x):=P(X in {x})$, where $X$ is the random variable that actually has exponential distribution to parameter $lambda$, rather than the density function $f$.
I need clarity on the terms, and perhaps an explanation on how it all fits together.
probability-theory probability-distributions
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
$endgroup$
After slowly coming to grips with Random Variables and the distributions that they subscribe to (e.g. $X$ ~ $mathcal{U}(0,1$)) , we introduced the notion of pdf, which I believe I understand in essence, but I am rather confused when, for example, we are told that for the exponential distribution to parameter $lambda > 0$
the density $f$ can be said to be $f(x):=lambda e^{-lambda x}chi_{[0,infty[}(x)$.
I always thought that distributions were set according to a random variable. However, a random variable is not mentioned anywhere above.
My guess is that $f(x):=P(X in {x})$, where $X$ is the random variable that actually has exponential distribution to parameter $lambda$, rather than the density function $f$.
I need clarity on the terms, and perhaps an explanation on how it all fits together.
probability-theory probability-distributions
probability-theory probability-distributions
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
asked Dec 5 '18 at 18:12
SABOYSABOY
641311
641311
1
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
1
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54
|
show 1 more comment
1
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
1
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54
1
1
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
1
1
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54
|
show 1 more comment
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1
$begingroup$
A random variable has a distribution, $P[X le alpha]$. Many variables may have the same distribution (for example all dice have essentially the same distribution). The density is one way of specifying the distribution (not all distributions have a density). Given a distribution (it must satisfy certain properties) one can construct a random variable having that distribution.
$endgroup$
– copper.hat
Dec 5 '18 at 18:17
1
$begingroup$
The density $f$ is not a probability; it is probability per unit of something. $f$ is the density of the random variable $X$, which as you say, has an exponential distribution.
$endgroup$
– StubbornAtom
Dec 5 '18 at 18:46
$begingroup$
@copper.hat what distributions do not have a density? discrete distributions?
$endgroup$
– SABOY
Dec 5 '18 at 19:04
$begingroup$
Any discrete distribution (such as a coin toss). Also, a distribution may be, for example, a mixture of discrete and continuous.
$endgroup$
– copper.hat
Dec 5 '18 at 19:06
$begingroup$
Strict terminology Distribution function, $P(Xle x)=lambdaint_0^xe^{-lambda u}du=1-e^{-lambda x}$ for $xge 0$.
$endgroup$
– herb steinberg
Dec 5 '18 at 19:54