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The probability density function of a random variable X is the exponential distribution of parameter $\frac{1}{2}$ . The probability density function of a random variable Y is a standard normal distribution. X and Y are independent of each other.

What is the probability density function of $Z=(2X−1)Y$ ?

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    $\begingroup$ What have you tried? $\endgroup$
    – Dominik Kutek
    Sep 21, 2019 at 20:03
  • $\begingroup$ I tried a change of variable $(z,t)=((2X-1)Y, 2X-1)$ but I don't have a converging integral when I try to calculate . $\endgroup$
    – user159729
    Sep 21, 2019 at 21:30
  • $\begingroup$ You could find the density of $W:=2X-1$ and then use the product density formula: $$ f_Z(z) = \int_{\mathbb R}f_W(w)f_Y(z/w)\frac1{|w|}\ \mathsf dw, $$ although that integral may be difficult to compute. $\endgroup$
    – Math1000
    Sep 21, 2019 at 21:47
  • $\begingroup$ I tried this but I have a problem finding an integral that converges.What result should this formula give ? $\endgroup$
    – user159729
    Sep 21, 2019 at 22:04

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