Therefore in either case p c 2c so that is itself

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Unformatted text preview: OM VARIABLE 99 The above three step procedure addresses the case that Y is continuous-type. If the function g is piecewise constant, such as a quantizer function, then Y is a discrete-type random variable. In that case, Steps 2 and 3 should be replaced by the following single step: Step 2: (if Y is discrete-type) 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.

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