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I am given the following PDF of random variable $X$:

$$f(x)= \begin{cases} e^x & \text{for }x<0, \\ 0 & \text{otherwise}. \end{cases}$$

a) Compute $E(e^x)$:

Here is my work:

$$E(e^x)=\int_{-\infty}^0 xe^x dx = -1$$

b) Find PDF of $Y=X-2$:

I am really stuck on how to do this part

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closed as off-topic by Did, Edward Jiang, Daniel W. Farlow, Leucippus, k170 Apr 27 '16 at 1:15

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  • $\begingroup$ Look here under "Dependent variables and change of variables": en.m.wikipedia.org/wiki/Probability_density_function $\endgroup$ – Theodor Johnson Apr 26 '16 at 18:34
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    $\begingroup$ How do you pass from f(x)=e^(x) for x<0 f(x)=0 otherwise to E(e^(x))=\int_{-inf}^0 xe^(x) dx? $\endgroup$ – Did Apr 26 '16 at 18:35
  • $\begingroup$ E(e^(x)) = integral from negative infinity to 0 of [xe^(x)]dx = e^(x)(x-1) evaluate from negative infinity to 0 = (1)(-1)-(0) = -1 $\endgroup$ – Lcheck Apr 26 '16 at 19:56
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a) There is a formula $$E[g(X)] = \int_S g(x)f_X(x)\,dx$$ where $S$ is the support of a random variable $X$, and $g(X)$ is a function of that random variable. It looks like you want to compute $$E[e^X] = \int_{-\infty}^0 e^xf_X(x)\,dx = \int_{-\infty}^0 e^x\cdot e^x\,dx.$$

b) The popular method is to compute $$P(Y\leq y) = P(X-2\leq y) = P(X\leq y+2)$$ to find the CDF of $Y$.

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  • $\begingroup$ For part A I got (1/2), does that sound correct? Also for part B I am not sure where to go from there $\endgroup$ – Lcheck Apr 26 '16 at 20:00
  • $\begingroup$ Do you know what $P(X\leq x)$ means? This isn't much different. $\endgroup$ – Em. Apr 26 '16 at 21:11
  • $\begingroup$ I know I've seen it before, I understand the similarity. But I'm not sure what that means I need to calculate $\endgroup$ – Lcheck Apr 26 '16 at 23:42
  • $\begingroup$ Loosely speaking, it means the area under the curve up to $x$. In our case, we have $P(X\leq x) = \int_{-\infty}^x f_X(t)\,dt$ for $x\leq 0$. $\endgroup$ – Em. Apr 26 '16 at 23:56
  • $\begingroup$ Okay, so I would need to take the integral from negative infinity to (x-2) and that would give me the CDF, correct? And from there how do I get to the PDF? $\endgroup$ – Lcheck Apr 27 '16 at 0:55
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In general when you are looking for the pdf of a random variable of the kind

$Y = aX + b$

where $a$ and $b$ are real constants you use the standard transformation of variables formula with $x = g^{-1}(y) = h(y)$

$$f_Y(y) = f_X(x)\left|\frac{dx}{dy}\right| = f_X[h(y)]\left|\frac{dh(y)}{dy}\right|$$

Applying this to $Y = aX + b$ you get

$$f_Y(y) = \frac{1}{|a|}f_X \left( \frac{y-b}{a} \right)$$

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