Variance and covariance inequalityMinimizing the variance of weighted sum of two random variables with respect to the weightsFinding covariance given varianceVariance and covarianceVariance, Covariance, and Correlation answer checkConditional Expectation and Variance QuestionVariance and covariance between 2 variablesHow to prove an equality envolving variance and covarianceFinding the covariance of X,Y using variance?Covariance between given iid random variable and sample averageVariance of ratio of mean value of functions of a random variable
Why commonly or frequently used fonts sizes are even numbers like 10px, 12px, 16px, 24px, or 32px?
With today's technology, could iron be smelted at La Rinconada?
Is there any deeper thematic meaning to the white horse that Arya finds in The Bells (S08E05)?
Were any toxic metals used in the International Space Station?
Did any "washouts" of the Mercury program eventually become astronauts?
Is there an academic word that means "to split hairs over"?
Meaning of "legitimate" in Carl Jung's quote "Neurosis is always a substitute for legitimate suffering."
Wireless headphones interfere with Wi-Fi signal on laptop
Does it matter what way the tires go if no directional arrow?
Why did Varys remove his rings?
How did the horses get to space?
How to rename multiple files in a directory at the same time
Promotion comes with unexpected 24/7/365 on-call
Why would company (decision makers) wait for someone to retire, rather than lay them off, when their role is no longer needed?
What is the effect of the Feeblemind spell on Ability Score Improvements?
Why do OOK transmissions have bandwidth?
How does Ctrl+c and Ctrl+v work?
How could it be that 80% of townspeople were farmers during the Edo period in Japan?
What do the "optional" resistor and capacitor do in this circuit?
tikzcd diagram within an array
Understanding Python syntax in lists vs series
Can my American children re-enter the USA by International flight with a passport card? Being that their passport book has expired
Why is Drogon so much better in battle than Rhaegal and Viserion?
Assembly writer vs compiler
Variance and covariance inequality
Minimizing the variance of weighted sum of two random variables with respect to the weightsFinding covariance given varianceVariance and covarianceVariance, Covariance, and Correlation answer checkConditional Expectation and Variance QuestionVariance and covariance between 2 variablesHow to prove an equality envolving variance and covarianceFinding the covariance of X,Y using variance?Covariance between given iid random variable and sample averageVariance of ratio of mean value of functions of a random variable
$begingroup$
Given a real-valued random variable $X$, is
$$2mathbb E[X] mathrmVar(X) geq mathrmCov(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
probability inequality variance covariance
asked 4 hours ago
sk1ll3rsk1ll3r
405
$endgroup$
add a comment |
$begingroup$
Given a real-valued random variable $X$, is
$$2mathbb E[X] mathrmVar(X) geq mathrmCov(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
probability inequality variance covariance
asked 4 hours ago
sk1ll3rsk1ll3r
405
$endgroup$
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago
add a comment |
$begingroup$
Given a real-valued random variable $X$, is
$$2mathbb E[X] mathrmVar(X) geq mathrmCov(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
probability inequality variance covariance
asked 4 hours ago
sk1ll3rsk1ll3r
405
$endgroup$
Given a real-valued random variable $X$, is
$$2mathbb E[X] mathrmVar(X) geq mathrmCov(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
probability inequality variance covariance
probability inequality variance covariance
asked 4 hours ago
sk1ll3rsk1ll3r
405
asked 4 hours ago
sk1ll3rsk1ll3r
405
asked 4 hours ago
sk1ll3rsk1ll3r
405
asked 4 hours ago
sk1ll3rsk1ll3r
405
asked 4 hours ago
sk1ll3rsk1ll3r
405
405
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago
add a comment |
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Use the variance and covariance identities
$$textVar(X)=mathbb E[X^2] − mathbb E[X]^2$$
and
$$textCov(X,Y) = mathbb E[XY] − mathbb E[X] mathbb E[Y]$$
Then the given inequality is equivalent to
$$ 2 mathbb E[X] Big( mathbb E[X^2] - mathbb E[X]^2 Big)
overset?geq
mathbb E[X^3] - mathbb E[X] mathbb E[X^2] tag1$$
For a counterexample, let $X sim textExp(lambda)$ be the exponential distribution with parameter $lambda in (0, infty)$ and probability mass function
$$ f_X(x) = begincases
lambda e^-lambda x & textif x geq 0 \
0 & textif x < 0
endcases $$
Its $n$-th moment is given by $mathbb E[X^n]=fracn!lambda^n$, so the inequality $(1)$ becomes
$$2 cdot frac1lambda Big( frac2lambda^2 - frac1lambda^2 Big)
geq
frac6lambda^3 - frac1lambda cdot frac2lambda^2$$
which reduces to
$$frac2lambda^3 geq frac4lambda^3$$
This is false for all $lambda in (0, infty)$, hence we found a counterexample.
answered 3 hours ago
M. NestorM. Nestor
880115
$endgroup$
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
add a comment |
$begingroup$
Alternate approach / observation...
Assume $2 E[X] Var(X) ge Cov(X, X^2)$ for any real-valued r.v. $X.$ Then in particular it would also be true for $-X$, so that:
$$
beginalign
2E[-X] Var(-X) &ge Cov(-X, (-X)^2)\
-2E[X] Var(X) &ge -Cov(X, X^2)\
2E[X] Var(X) &le Cov(X, X^2)
endalign
$$
which, combined with the original inequality, means: $2E[X]Var(X) = Cov(X, X^2)$ for all $X$.
At this point, it would seem one should be able to find a counter-example easily, without using e.g. $Cov(X, X^2) = E[X^3] - E[X]E[X^2]$ etc, but I just couldn't think of a quick one. :(
answered 43 mins ago
antkamantkam
4,479415
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3226379%2fvariance-and-covariance-inequality%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Use the variance and covariance identities
$$textVar(X)=mathbb E[X^2] − mathbb E[X]^2$$
and
$$textCov(X,Y) = mathbb E[XY] − mathbb E[X] mathbb E[Y]$$
Then the given inequality is equivalent to
$$ 2 mathbb E[X] Big( mathbb E[X^2] - mathbb E[X]^2 Big)
overset?geq
mathbb E[X^3] - mathbb E[X] mathbb E[X^2] tag1$$
For a counterexample, let $X sim textExp(lambda)$ be the exponential distribution with parameter $lambda in (0, infty)$ and probability mass function
$$ f_X(x) = begincases
lambda e^-lambda x & textif x geq 0 \
0 & textif x < 0
endcases $$
Its $n$-th moment is given by $mathbb E[X^n]=fracn!lambda^n$, so the inequality $(1)$ becomes
$$2 cdot frac1lambda Big( frac2lambda^2 - frac1lambda^2 Big)
geq
frac6lambda^3 - frac1lambda cdot frac2lambda^2$$
which reduces to
$$frac2lambda^3 geq frac4lambda^3$$
This is false for all $lambda in (0, infty)$, hence we found a counterexample.
answered 3 hours ago
M. NestorM. Nestor
880115
$endgroup$
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
add a comment |
$begingroup$
Use the variance and covariance identities
$$textVar(X)=mathbb E[X^2] − mathbb E[X]^2$$
and
$$textCov(X,Y) = mathbb E[XY] − mathbb E[X] mathbb E[Y]$$
Then the given inequality is equivalent to
$$ 2 mathbb E[X] Big( mathbb E[X^2] - mathbb E[X]^2 Big)
overset?geq
mathbb E[X^3] - mathbb E[X] mathbb E[X^2] tag1$$
For a counterexample, let $X sim textExp(lambda)$ be the exponential distribution with parameter $lambda in (0, infty)$ and probability mass function
$$ f_X(x) = begincases
lambda e^-lambda x & textif x geq 0 \
0 & textif x < 0
endcases $$
Its $n$-th moment is given by $mathbb E[X^n]=fracn!lambda^n$, so the inequality $(1)$ becomes
$$2 cdot frac1lambda Big( frac2lambda^2 - frac1lambda^2 Big)
geq
frac6lambda^3 - frac1lambda cdot frac2lambda^2$$
which reduces to
$$frac2lambda^3 geq frac4lambda^3$$
This is false for all $lambda in (0, infty)$, hence we found a counterexample.
answered 3 hours ago
M. NestorM. Nestor
880115
$endgroup$
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
add a comment |
$begingroup$
Use the variance and covariance identities
$$textVar(X)=mathbb E[X^2] − mathbb E[X]^2$$
and
$$textCov(X,Y) = mathbb E[XY] − mathbb E[X] mathbb E[Y]$$
Then the given inequality is equivalent to
$$ 2 mathbb E[X] Big( mathbb E[X^2] - mathbb E[X]^2 Big)
overset?geq
mathbb E[X^3] - mathbb E[X] mathbb E[X^2] tag1$$
For a counterexample, let $X sim textExp(lambda)$ be the exponential distribution with parameter $lambda in (0, infty)$ and probability mass function
$$ f_X(x) = begincases
lambda e^-lambda x & textif x geq 0 \
0 & textif x < 0
endcases $$
Its $n$-th moment is given by $mathbb E[X^n]=fracn!lambda^n$, so the inequality $(1)$ becomes
$$2 cdot frac1lambda Big( frac2lambda^2 - frac1lambda^2 Big)
geq
frac6lambda^3 - frac1lambda cdot frac2lambda^2$$
which reduces to
$$frac2lambda^3 geq frac4lambda^3$$
This is false for all $lambda in (0, infty)$, hence we found a counterexample.
answered 3 hours ago
M. NestorM. Nestor
880115
$endgroup$
Use the variance and covariance identities
$$textVar(X)=mathbb E[X^2] − mathbb E[X]^2$$
and
$$textCov(X,Y) = mathbb E[XY] − mathbb E[X] mathbb E[Y]$$
Then the given inequality is equivalent to
$$ 2 mathbb E[X] Big( mathbb E[X^2] - mathbb E[X]^2 Big)
overset?geq
mathbb E[X^3] - mathbb E[X] mathbb E[X^2] tag1$$
For a counterexample, let $X sim textExp(lambda)$ be the exponential distribution with parameter $lambda in (0, infty)$ and probability mass function
$$ f_X(x) = begincases
lambda e^-lambda x & textif x geq 0 \
0 & textif x < 0
endcases $$
Its $n$-th moment is given by $mathbb E[X^n]=fracn!lambda^n$, so the inequality $(1)$ becomes
$$2 cdot frac1lambda Big( frac2lambda^2 - frac1lambda^2 Big)
geq
frac6lambda^3 - frac1lambda cdot frac2lambda^2$$
which reduces to
$$frac2lambda^3 geq frac4lambda^3$$
This is false for all $lambda in (0, infty)$, hence we found a counterexample.
answered 3 hours ago
M. NestorM. Nestor
880115
edited 1 hour ago
answered 3 hours ago
M. NestorM. Nestor
880115
answered 3 hours ago
M. NestorM. Nestor
880115
answered 3 hours ago
M. NestorM. Nestor
880115
880115
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
add a comment |
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
$begingroup$
hi! thanks a lot for this. do you think the inequality is false for a real-valued $X$ (as specified in the question) as well? i haven't been able to find a counter-example. for $X sim mathrmNormal(mu, sigma^2)$, both left and right hand side reduce to $2musigma^2$.
$endgroup$
– sk1ll3r
2 hours ago
1
1
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
$begingroup$
Sorry I did not notice that detail. I edited my answer to include an example where $X$ has continuous support.
$endgroup$
– M. Nestor
1 hour ago
add a comment |
$begingroup$
Alternate approach / observation...
Assume $2 E[X] Var(X) ge Cov(X, X^2)$ for any real-valued r.v. $X.$ Then in particular it would also be true for $-X$, so that:
$$
beginalign
2E[-X] Var(-X) &ge Cov(-X, (-X)^2)\
-2E[X] Var(X) &ge -Cov(X, X^2)\
2E[X] Var(X) &le Cov(X, X^2)
endalign
$$
which, combined with the original inequality, means: $2E[X]Var(X) = Cov(X, X^2)$ for all $X$.
At this point, it would seem one should be able to find a counter-example easily, without using e.g. $Cov(X, X^2) = E[X^3] - E[X]E[X^2]$ etc, but I just couldn't think of a quick one. :(
answered 43 mins ago
antkamantkam
4,479415
$endgroup$
add a comment |
$begingroup$
Alternate approach / observation...
Assume $2 E[X] Var(X) ge Cov(X, X^2)$ for any real-valued r.v. $X.$ Then in particular it would also be true for $-X$, so that:
$$
beginalign
2E[-X] Var(-X) &ge Cov(-X, (-X)^2)\
-2E[X] Var(X) &ge -Cov(X, X^2)\
2E[X] Var(X) &le Cov(X, X^2)
endalign
$$
which, combined with the original inequality, means: $2E[X]Var(X) = Cov(X, X^2)$ for all $X$.
At this point, it would seem one should be able to find a counter-example easily, without using e.g. $Cov(X, X^2) = E[X^3] - E[X]E[X^2]$ etc, but I just couldn't think of a quick one. :(
answered 43 mins ago
antkamantkam
4,479415
$endgroup$
add a comment |
$begingroup$
Alternate approach / observation...
Assume $2 E[X] Var(X) ge Cov(X, X^2)$ for any real-valued r.v. $X.$ Then in particular it would also be true for $-X$, so that:
$$
beginalign
2E[-X] Var(-X) &ge Cov(-X, (-X)^2)\
-2E[X] Var(X) &ge -Cov(X, X^2)\
2E[X] Var(X) &le Cov(X, X^2)
endalign
$$
which, combined with the original inequality, means: $2E[X]Var(X) = Cov(X, X^2)$ for all $X$.
At this point, it would seem one should be able to find a counter-example easily, without using e.g. $Cov(X, X^2) = E[X^3] - E[X]E[X^2]$ etc, but I just couldn't think of a quick one. :(
answered 43 mins ago
antkamantkam
4,479415
$endgroup$
Alternate approach / observation...
Assume $2 E[X] Var(X) ge Cov(X, X^2)$ for any real-valued r.v. $X.$ Then in particular it would also be true for $-X$, so that:
$$
beginalign
2E[-X] Var(-X) &ge Cov(-X, (-X)^2)\
-2E[X] Var(X) &ge -Cov(X, X^2)\
2E[X] Var(X) &le Cov(X, X^2)
endalign
$$
which, combined with the original inequality, means: $2E[X]Var(X) = Cov(X, X^2)$ for all $X$.
At this point, it would seem one should be able to find a counter-example easily, without using e.g. $Cov(X, X^2) = E[X^3] - E[X]E[X^2]$ etc, but I just couldn't think of a quick one. :(
answered 43 mins ago
antkamantkam
4,479415
answered 43 mins ago
antkamantkam
4,479415
answered 43 mins ago
antkamantkam
4,479415
answered 43 mins ago
antkamantkam
4,479415
4,479415
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3226379%2fvariance-and-covariance-inequality%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
One trick is to write a little computer program to generate examples randomly, which you can use to find counterexamples. Such as $P(X=a)=p, P(X=b)=1-p$, where you use a random number generator to pick the parameters $a,b,p$ and exact formulas in terms of $a,b.p$ for the relevant moments of $X$.
$endgroup$
– kimchi lover
4 hours ago
$begingroup$
I expect that the answer will follow quickly after applying the well-known identities $textVar(X)=mathbb E[X^2]-mathbb E[X]^2$ and $textCov(X,Y)=mathbb E[XY]-mathbb E[X]mathbb E[Y]$
$endgroup$
– M. Nestor
4 hours ago