This preview shows page 1. Sign up to view the full content.
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. CONTINUOUSTYPE 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 discretetype 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 < 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 < c < 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...
View
Full
Document
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

Click to edit the document details