Dear researchers
I have a random variable X that is following a bimodal distribution, when I squared each element in that variable X, I got a distribution shape similar to exponential or Chi-squared, Could you please advise me any reference or explanation on how to prove what's exactly the distribution in such case?
Best regards
If you could please clarify? A bimodal distribution takes on two values, say 0 and 1, and squaring those values would keep those values the same (as well as the frequencies associated with the squared values).
Having said this, perhaps it's helpful for you to know that a chi-square distribution is a sum of squared normal deviates (e.g., scores that equal the sum of squared z-scores across a set of variables).
Dear Frederick L Oswald
In my case, I have data that is following a bimodal distribution, When I squared this data and visualized its new distribution, I noticed it is not a bimodal any more, But it is rather an exponential or may be a chi-squared.
If the bimodal distribution is a mix of two normal distributions, then the squared bimodal is a mix of two gamma distributions.
Xbimodal = i * Xa + (1-i)*Xb
with i ~ Bernoulli(p), Xa ~ N(µa,s²a), Xb ~ N(µb,s²b)
(Xbimodal)² = i * (Xa)² + (1-i)*(Xb)²
=>
(X)² ~ Gamma( shape=µ²*rate + 1, rate=exp(-s)/2 )
Dear Jochen Wilhelm,
thanks very much for your valuable clarification.
Best,
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