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Unformatted text preview: OM VARIABLE 99 The above three step procedure addresses the case that Y is continuoustype. If the function g is
piecewise constant, such as a quantizer function, then Y is a discretetype random variable. In that
case, Steps 2 and 3 should be replaced by the following single step:
Step 2: (if Y is discretetype) Find the pmf of Y. Work with the deﬁnition of the pmf:
pY (v ) = P {Y = v } = P {g (X ) = v }. In order to ﬁnd the probability of the event {g (X ) = v },
try to describe it in a way that involves X in a simple way. Basically, P {g (X ) = v } =
u:g (u)=v pX (u). Here pY (v ) only needs to be identiﬁed for v in the support of Y. Usually the
pmf is the desired answer and there is no need to ﬁnd the CDF.
Example 3.8.1 Suppose Y = X 2 , where X is a random variable with pdf fX (u) =
Find the pdf, mean, and variance of Y.
Solution: e−u
2 for u ∈ R. Note that Y = g (X ), where g is the function g (u) = u2 . Sketches of fX and g are f (u)
X g(u) u Figure 3.13: The pdf of X and the function g .
shown in Figure 3.13. The support of fX is the entire real line, and g maps the real line onto the
nonn...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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