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This is a question which I got stuck for over an hour, please help..

Let $X$ be a random variable for which $E(X)=\mu$ and $Var(X)=\sigma^2$. Construct a probability distribution for $X$ such that $$\Bbb{P}(|X-\mu|\ge 3\sigma)=\frac{1}{9}$$.

I know this is the upper bound which the above probability can take (according to Chebyshev's inequality). But I don't know how to make use of this fact in finding a distribution. I tried a normal distribution but it wasn't right.

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  • $\begingroup$ It sounds from your question like after your initial thought (normal), which would take seconds to check even if it wasn't obvious, you tried nothing whatever.. What did you do with the other 59 minutes? Trying only the first thing to come into your head is not really the way to get anywhere. The benefit comes from wrestling with these things a bit. $\endgroup$ – Glen_b Dec 31 '15 at 6:32
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    $\begingroup$ Please add the self-study tag, read its tag-wiki and modify your question to follow the guidelines on asking such questions. In particular, you'll need to clearly identify what you've done to solve the problem yourself, and indicate the specific help you need at the point you struck difficulty. $\endgroup$ – Glen_b Dec 31 '15 at 6:41
  • $\begingroup$ I must mention here that I am a novice on this subject. I first, starting from too much basics, tried to integrate the pdf of $N~(\mu ,\sigma^2)$ from $\mu -3\sigma$ to $\mu + 3\sigma$. Then I realised it's not easily integrable, and then it clicked on my mind that why not standardise to $N~(0,1)$. Then I found, I didn't have a Normal distribution table at hand, so I looked up the internet and found one. All this took the time..... It wasn't much fruitful work, so I didn't mention this in my question.. @Glen_b $\endgroup$ – Qwerty Dec 31 '15 at 7:54
  • $\begingroup$ An examination of any proof of Chebyshev's Inequality will suggest a solution, because such proofs clearly exhibit the way in which the inequality generally fails to an equality. $\endgroup$ – whuber Dec 31 '15 at 14:28
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  1. What happens if you try a Bernoulli($p$)? Try $p$ small (say $p=0.05$). How much is outside the bound? How big can you make $p$ and still have something outside the bound?

  2. Can you construct a similar, symmetric example that does better?

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  • $\begingroup$ I am still at a loss as how to proceed with Bernoulli distribution. For normal, or others,I have a pdf to integrate over, what am I to do with this?. Please forgive me if I have needed up with the basics.... @Glen_b $\endgroup$ – Qwerty Dec 31 '15 at 8:33
  • $\begingroup$ No wonder you couldn't do it! If you don't know how probability functions and probability calculations work with simple discrete distributions, you're definitely not in a position to attempt this question; you need to review the basics of discrete distributions first. You need to be able to work out the mean and standard deviation of the Bernoulli as well as work out the proportion past $x$ (which for a Bernoulli is trivial) $\endgroup$ – Glen_b Dec 31 '15 at 9:58
  • $\begingroup$ One more doubt, for this question, do I have to check for all distributions which one is/are working or not,(trial and error process) or is there some other determining process? $\endgroup$ – Qwerty Dec 31 '15 at 12:41

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