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Let $\eta$ be a random variable that takes the value $0.5$ or $-0.5$ with equal probability $0.5$..Define $\theta_n= \dfrac{\sum_{i=1}^n \eta_i}{\sqrt{n}},$ where $\eta_i$ are independent variables with the same distribution as $\eta.$ Find the values that $\theta$ can take and their probabilities for $n= 3,6,9.$ Plot their histograms together with the pdf of the limit of $\theta \text{ as } n \rightarrow \infty.$

The plots I got: enter image description here

The problem is when I choose a very big $n,$ $\theta$ doesn't seem to converge to any value I just keep getting random values at each try. enter image description here

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  • $\begingroup$ Are you sure the second plot is OK? The $y$ axis seems to be counting about 10 trials. Also, I think you should be plotting the $\theta_n$‘s, not the $\eta_n$s right? $\endgroup$
    – Fede Poncio
    Commented Oct 11, 2020 at 17:34
  • $\begingroup$ Well, the second plot should show densities by passing density=True as an argument to the plot function. I'm not sure about plotting $\theta_n$ $\endgroup$
    – cosine
    Commented Oct 11, 2020 at 17:40

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