[Math] Convolution of uniform PDF and normal PDF in Matlab

Question

I have a problem with Matlab and probability density functions:

$x$ is a random variable, uniformly distributed between $[-0.5,0.5]$, so its probability density function is $p_x(t)=\text{rect}(t)$.

$w$ is also a random variable, randomly distributed with zero-mean and variance of 1. $p_w(t)=\frac{1}{\sqrt{2\pi}}\exp(\frac{-t^2}{2})$.

$x$ and $w$ are uncorrelated.

I have to find $p_{x+w}(t)$ and I know that this is equal to the convolution between $p_x(t)$ and $p_w(t)$:
$$
\int\frac{1}{\sqrt{2\pi}}\exp\Big(\frac{-(t-\tau)^2}{2}\Big)\text{rect}(\tau)d\tau = \int_{-0.5}^{0.5} \frac{1}{\sqrt{2\pi}}\exp\Big(\frac{-(t-\tau)^2}{2}\Big)d\tau \\= -\int_{t-0.5}^{t+0.5} \frac{1}{\sqrt{2\pi}}\exp\Big(\frac{-z^2}{2}\Big)dz = Q(t-0.5)-Q(t+0.5)
$$

My problem occurs when I plot this result with matlab:

plot(t,qfunc(t-0.5)-qfunc(t+0.5));

It's different from plotting the convolution of the two PDF's:

plot(t,decimate(conv(rectpuls(t),normpdf(t,0,1)),2));

Is there a way to understand which one of the two plots is the correct one?

Thank you.

Answer 1

Comment:

Let $U \sim \mathsf{Unif}(-.5, .5)$ and $Z \sim \mathsf{Norm}(0,1).$ Then $X = U + Z.$ I suppose that in your final result you use $Q$ for the standard normal CDF, often written $\Phi.$ However, what you have is a non-positive function. I think it should be $\Phi(x + .5)-\Phi(x - .5).$ [Thus, your analysis is correct except for a simple mistake in algebra.]

The following demonstration in R statistical software with a million observations $X$ shows a histogram consistent with that density function:

 u = runif(10^6, -.5, .5);  z = rnorm(10^6);  x = u + z
 hist(x, prob=T, br=50, col="skyblue2")
    curve(pnorm(x+.5)-pnorm(x-.5), col="red", lwd=2, add=T)

Note: In R runif and rnorm sample from uniform and normal distributions and pnorm is a normal CDF. I'm sorry not to use Matlab, but I do not have access to it.