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Consider a Probability density function f(x) which is uniformly distributed between say (-5,5).Now if i define a random variable "y" which is related to the random variable "x" as follows:

y = 1 ; -2.5 < x < 2.5. (zero outside)

Since this mapping neither one to one nor does it correspond to a monotonic function.

How can i find the probability density function of transformed random variable?

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  • $\begingroup$ Our site offers a wealth of worked examples and techniques for solving such problems. Why not search it? One of the higher-voted threads that turns up looks like it fully answers your question: see stats.stackexchange.com/questions/144138. Please note that your example is not appropriate for your question, because $Y$ will not have a PDF at all. $\endgroup$
    – whuber ♦
    Commented Jan 13, 2016 at 0:28

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The above type of random variable transformation requires the use of dirac delta distribution as explained in RV transformation

The general form is:

enter image description here

Here f(x)= 1 for -2.5 < x < 2.5 and f(x)=0 outside.

Thus,

fy(y) = 0.5 delta(y-1) + 0.5 delta(y).

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