So the question is to discover the median of $X$ if the mode of the circulation is at $x = sqrt2/4$. And the random variable $X$ has the density function

$$f(x) = left{ eginarrayll kx & mboxfor $0 le x le sqrtfrac2k$ ;\ 0 & mboxotherwise.endarray ideal.$$

I understand that the median of a PDF is such that the integral is equated to half. If I proceed that way via this trouble, I am acquiring a response of $1$. But the answer is offered is $frac14$. Also, I am not sure why the mode is offered. If I equate $f"(x)$ to $sqrt2/4$ (which is the given mode), I end up obtaining random options.

Can someone help me out right here, please?


$egingroup$ Siong Thye Goh has actually currently gave a nice explacountry for your question. The mode is offered simply to discover the worth of $k$. The p.d.f of $X$ is a raising attribute (in fact a straight line via slope $k>0$) in the interval $left<0, sqrtfrac2k ight>$. Since mode exists it need to be at the finish suggest of this interval. This gives you $x=fracsqrt24=sqrtfrac2k$ $endgroup$
You are told the mode so that we recognize the value of $k$.

You are watching: How to find the median of a pdf


Hence $k = 16.$

$$int_0^m 16x , dx = frac12$$

Try to deal with for $m$ from the above equation.


The mode is in this case $sqrt2/k$ so the reality that it likewise equalizes $sqrt2/4$ tells us that $k=16$.

Now we have the right to discover the median $m$ on base of $$int^m_016xdx=frac12$$causing $$m=frac14$$


Thanks for contributing a solution to soimg.orgematics Stack Exchange!

Please be certain to answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based upon opinion; ago them up with references or individual endure.

Use soimg.orgJax to format equations. soimg.orgJax reference.

See more: Why Does Willy Tell Howard About Dave Singleman ?​ Death Of A Salesman

To learn more, check out our tips on writing good answers.

Post Your Answer Discard

By clicking “Blog post Your Answer”, you agree to our regards to business, privacy policy and cookie plan

Not the answer you're looking for? Browse other questions tagged probability-distributions random-variables or ask your very own question.


Given is a uniformly dispersed random variable $u in (0,1)$ and density. How deserve to you create a random variable through that density?

site architecture / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.9.2.40142

Your privacy

By clicking “Accept all cookies”, you agree Stack Exreadjust have the right to keep cookies on your tool and discshed information in accordance via our Cookie Policy.