Warmup Problem for Problem Set 5 (18.440)
For the following examples, find the cdf FY of Y in terms of the cdf FX of X, and the pdf fY of Y in terms
of the pdf fX of X.
Hint: It may help to draw the plot of y as a function of x.
√
1. Y = X, where X > 0.
Hint: What happens to the very small interval [x, x + ] when we take the square-root of every number
in the interval? By what factor does it grow or shrink (approximately, for small )?
2. Y = X 2 , where X can take any value in R.
Hint (cdf): For what values of x is x2 < y?
Hint (pdf): For what values of x is x2 = y?
3. The piecewise function
Y = X if X ≤ 1
Y = 1 if 1 < X ≤ 2
Y = X − 1 if X > 2.
Hint (cdf): The cdf is defined as FY (y) := P (Y ≤ y), NOT P (Y < y).
Hint (pdf): At what points is the pdf not a function (technically it’s a "delta functional") in this
example?
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