Math Question Help (1 Viewer)

[frankfurt]

Hi Guys,

I came across this question that I'm struggling with:


Please help.

Thanks in advance!

 

[cossine]

Hi Guys,

I came across this question that I'm struggling with:
View attachment 35815

Please help.

Thanks in advance!

The theorem mention is very similar to Markov's Inequality, however the integral on the right-hand-side has a lower-bound of x instead of zero.

I think it might be interesting to see if the theorem is actually true so maybe test some probability density function on it such as exponential or beta distribution.

Edit:

The theorem is true just take a look at the proof of Markov Inequality, "Probability Theoretic Approach": "Method 1". You should be able to use part of the proof.

https://en.wikipedia.org/wiki/Markov's_inequality

 

[frankfurt]

The theorem mention is very similar to Markov's Inequality, however the integral on the right-hand-side has a lower-bound of x instead of zero.

I think it might be interesting to see if the theorem is actually true so maybe test some probability density function on it such as exponential or beta distribution.

Edit:

The theorem is true just take a look at the proof of Markov Inequality, "Probability Theoretic Approach": "Method 1". You should be able to use part of the proof.

https://en.wikipedia.org/wiki/Markov's_inequality

Yeah I had a similar thought to use Markov's inequality. However, it's difficult to get rid of the integral of xf(x) from 0 to a in the example above. How would you go about doing that?

 

[cossine]

Yeah I had a similar thought to use Markov's inequality. However, it's difficult to get rid of the integral of xf(x) from 0 to a in the example above. How would you go about doing that?

So if take a look at the proof that should give you insight. There is chain of inequalities in proof 1. You should find integral of xf(x) from x to infinity there.