What is the pdf and cdf of $Y := XD +(1-X)P$ where $X \sim Ber(p)$, $D\sim Exp(\lambda_d)$,$P\sim Exp(\lambda_p)$?

Question

What is the pdf and cdf of $Y := XD +(1-X)P$ where $X \sim Ber(p)$, $D\sim Exp(\lambda_d)$,$P\sim Exp(\lambda_p)$? All variables are independet.

Thanks for the hint. By the Hint below:

$$f_Y(y)=f_Y(y)\biggr\vert_{X=1}\Pr(X=1)+f_Y(y)\biggr\vert_{X=0}\Pr(X=0) = f_D(y)p + f_P(y)(1-p) = \lambda_d e^{-\lambda_d y}p + (1-p)\lambda_p e^{-\lambda_p y} $$

Is this correct?

Yes, this is called a mixture distribution. – msm Jun 17 '17 at 08:48 — msm, Jun 17 '17 at 08:48

score 1 · Accepted Answer · answered Jun 17 '17 at 08:16

1

Hint

$$f_Y(y)=f_Y(y)\biggr\vert_{X=1}\Pr(X=1)+f_Y(y)\biggr\vert_{X=0}\Pr(X=0)$$

answered Jun 17 '17 at 08:16

msm

7,147

What is the pdf and cdf of $Y := XD +(1-X)P$ where $X \sim Ber(p)$, $D\sim Exp(\lambda_d)$,$P\sim Exp(\lambda_p)$?

1 Answers1