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I want to do a hypothesis testing excersise that comes after, but i'm a little bit confused about the plot of the density of Z, which i feel i need to understand first:

enter image description here

More specifically, where do i see in the graph the "P[Y=1]=P[Y-1]=1/2" of the discrete RV? if i am adding a normal distribution to it, with a variance that reaches those limits in the x axis (-1,1).

How does the discrete RV affects my normal distribution in terms of limits?

Side note: I thought about using R to plot it as well, is it a good idea? or should i use matlab?

Thank you

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  • $\begingroup$ Do you mean plot the distribution function of $Z$? $\endgroup$ – StubbornAtom Jan 6 at 19:39
  • $\begingroup$ The density of Z $\endgroup$ – mjginno Jan 6 at 19:46
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I think you're asking for the density of $Z$. $Z=X+Y$ is either a normal RV with mean $1$ or mean $-1$ (variance is $1$), based on the value of $Y$. Let $f(x,\mu)$ be the density function of a normal RV with mean $\mu$ and variance $1$. Then, density of $Z$ is $f_Z(z)=f(z,1)/2 = f(z,-1)/2$. Recall that, $f(x,\mu)=\frac{1}{\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2}}$. When you plot this function, you'll see a two peak curve on your screen. R or Matlab doesn't matter.

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  • $\begingroup$ Thank you, that's much more clear, I understand better now. $\endgroup$ – mjginno Jan 7 at 15:13

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