Question

A random variable X is generated as follows. We flip a coin. With probability p, the result is Heads, and then X is generated according to a PDF fxh which is uniform on (0,1). With probability 1-p the result is Tails, and then X is generated according to a PDF fxt of the form fxT (2) = 2x, if x € [0,1]. (The PDF is zero everywhere else.) 1. What is the (unconditional) PDF fx () of X? For 0 < x < 1:

2. Calculate E[X].

2. The MAP rule decides in favor of Heads if X<a and in favor of Tails if X>a. What is a?

Answer 1

Co Axire(x) si it -te(0,77 to ow. /2x it at [011] ow. fxlt (x) - = inconditional pdf fx(x) of x. fx) = pfxth (x) + (1-P) fx 17 ) = pxl + (1-P) (22) p+l-2p)22 = p + 2x 2px i p/1-22)+22 XE[011) + x ( [o (1] (2) E[x] = p E [x] + (1-P) E [x2] where x ~ fxth and f ~ fylt. E[XL] = E [unit fol]] = Efxz) = T127) de Tea 0 To F [x]: pet + (1-P)?

No. Date : 1 1 P ( Tails (X= 14) = fxl7 () x (1-2) fxt (%) * 11-p] + fxlu cu) x (2x 44 x (1-2) (R$ %)* C1-P) + (1 Jxp so IP (Tails (X= y) = 1-P ITP TPC Tails X=x) fxl7 (2)x(1-P) fxLt (%)* 6-2) + fx 1 ()xp. - 2x(1-P) 20 (1-P)+ Pa 2x (1-D 22-22 p + P %

No. Date : , so PlHeads X= x) 1 - 2201-P) 22(1-) + P 2x (1-P) tp., so it 2x(1-P) > then decide in favow of tails otherwise head (t.e., when Pitails (X= x) P (Heads(x2) 2x (1-P)>p 212) so as P 2012