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Suppose I have $X\sim Uniform[a,b]$ and $Y\sim normal(0,d^2)$, what's the expression for the density of $Z=X+Y$?

Let $F_{Z}(z)$ be the cdf of $Z$ evaluated at $z$, and let $\Phi(\cdot)$ and $\phi$ be standard normal cdf and pdf respectively. I got

$F_{Z}(z)=\frac{1}{b-a}\int_{a}^{b}\Phi(\frac{z-x}{d})dx$,

differentiate wrt to $z$ on both sides gives

$f_{Z}(z)=\frac{1}{b-a}\int_{a}^{b}\phi(\frac{z-x}{d})\frac{1}{d}dx=\frac{1}{b-a}(\Phi(\frac{z-a}{d})-\Phi(\frac{z-b}{d}))$ .

Does this look correct?Thanks!

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  • $\begingroup$ What progress have you made with your homework problem? Please add this to the question in an edit, add the self-study tag when you edit, and read its wiki. $\endgroup$
    – Dave
    Aug 16 '20 at 21:59
  • $\begingroup$ @Dave Thanks, it's done! $\endgroup$
    – T34driver
    Aug 16 '20 at 22:50
  • $\begingroup$ I see two mistakes. 1) You don’t seem to be using the uniform $X$. 2) You solve this kind of problem using convolution. 3) Please add the self-study tag. $\endgroup$
    – Dave
    Aug 16 '20 at 23:03
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    $\begingroup$ The left side of your last displayed equation is a function of $z$ while the right side does not depend on $z$ at all. $\endgroup$ Aug 17 '20 at 1:14
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    $\begingroup$ Some posts on site address the convolution of uniform and normal. $\endgroup$
    – Glen_b
    Aug 17 '20 at 1:42
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Comment:

As a reality check here is a simulation for the convolution of $U \sim \mathsf{Unif}(a=2, b=7)$ and $Z \sim \mathsf{Norm}(\mu = 0, \sigma = 3).$

Thus $E(U+Z) = 4.5 + 0 = 4.5$ and $V(U+Z) = 25/12 +9 = 4.0833.$

set.seed(2020)
a = 2;  b = 7;  sg = 3
u = runif(10^6, a, b)
z = rnorm(10^6, 0, sg)
x = u + z
mean(x); mean(u);  mean(z);  mean(u) + mean(z)
[1] 4.497167        # aprx E(X) = 4.5
[1] 4.500343        # aprx E(U) = 4.5
[1] -0.003175144    # aprx E(Z) = 0
[1] 4.497167
var(x); var(u); 25/12; var(z); var(u) + var(u)
[1] 11.08561        # aprx Var(X)
[1] 2.081356        # aprx Var(U) = 25/12
[1] 2.083333        # 25/12
[1] 9.011073
[1] 4.162712

hist(x, prob=T, br=50, col="skyblue2", 
 main="Simulated Density of X")
curve(1/(b-a)*( pnorm((x-a)/sg) - pnorm((x-b)/sg) ),
  add=T, col="red", lwd=2)

enter image description here

Note: Figure revised after edit and comment from OP.

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  • $\begingroup$ Thanks! This is very helpful. I guess you entered the wrong formula for the red curve. It should be dnorm((x-a)/sg) - dnorm((x-b)/sg). $\endgroup$
    – T34driver
    Aug 17 '20 at 4:45
  • $\begingroup$ See this post, when a=0,b=1, sg=1, my formula agrees with the answer given there. stats.stackexchange.com/questions/365601/… $\endgroup$
    – T34driver
    Aug 17 '20 at 4:49
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    $\begingroup$ Thanks! My first comment was incorrect (I'm not familiar with r and thought your dnorm means distribution function), you need to use: pnorm((x-a)/sg) - pnorm((x-b)/sg). That is, first swap your a and b, a first b second, and then change your density function to distribution function. Then I guess it should work. $\endgroup$
    – T34driver
    Aug 17 '20 at 5:19
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    $\begingroup$ Deleted my previous comment to avoid confusion. Also, edited my Answer to use corrected PDF, which looks fine. Glad this is resolved. $\endgroup$
    – BruceET
    Aug 17 '20 at 5:36

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