SECTION 6.2 Continuous Random Variables 353
differentiating both sides with respect totand applying the Fundamen-
tal Theorem of Calculus, we get
fY(t) = 2t.Of course, this is the density function given a few pages earlier. In
summary, the square root of the uniform random-number generator
has a linear density function given byf(t) = 2t.
Assume, more generally, that we wish to transform data from the
random-number generator X = randso as to produce a new random
variableY having a given distribution functionfY. If we denote this
transformation byY =g(X), we have
∫t
−∞
fY(x)dx=P(Y ≤t) =P(g(X)≤t) =P(X≤g−^1 (t)) =g−^1 (t),which determinesg−^1 and hence the transformationg.
Exercises
- If X is the random variable having the triangular density curve
depicted on page 349, compute
(a)P(X≤ 1 /3)
(b)P(X≥ 2 /3)
(c)P(1/ 3 ≤X≤ 2 /3)
(d)P(X > .5)- Suppose that you perform an experiment where you invoke the
random-number generator twice and letZ be the sum of the two
random numbers.
(a) ComputeP(. 5 ≤Z≤ 1 .65) theoretically.