# Then repeat this procedure using the third and fourth

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Unformatted text preview: Y = X , which is the integer part of X , and let R = X − X , which is the remainder. Describe the distributions of Y and R, and ﬁnd the limit of the pdf of R as λ → 0. 106 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES (a) f !2 !1 0 X 1 2 (b) !2 !1 0 1 2 !2 !1 0 1 2 !2 !1 0 1 2 !2 !1 0 1 2 (c) (d) f !2 !1 0 Y 1 2 Figure 3.19: Geometric interpretation for pdf of |X |. Solution: Clearly Y is a discrete-type random variable with possible values 0, 1, 2, . . . , so it is suﬃcient to ﬁnd the pmf of Y . For integers k ≥ 0, k+1 pY (k ) = P {k ≤ X &lt; k + 1} = λe−λu du = e−λk (1 − e−λ ), k and pY (k ) = 0 for other k . Turn next to the distribution of R. Note that R = g (X ), where g is the function sketched in Figure 3.20. Since R takes values in the interval [0, 1], we shall let 0 &lt; c &lt; 1 and ﬁnd FR (c) = g(u) 1 c u 0 c 1 1+c 2 2+c 3 ... 3+c Figure 3.20: Function g such that R = g (X ). P {R ≤ c}. The event {R ≤ c} is equivalent to the event that X falls into the union of intervals, 3.8. FUNCTIONS OF A RANDOM VARIA...
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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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