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We are given a sample of 7 proportions for the percentage of cloud cover recorded at a set time every day for a week, where the sample mean is $\bar{x}=0.51$ and the sample variance is $s=0.3277$. We are supposed to fit a Beta distribution to this data using the Method of Moments and then calculate the approximate probability of more than 50% cloud cover using any numerical integration method.

Using the Beta distribution's mean and variance formulas with MoM I calculated the parameter estimates as $\alpha=0.676816428$ and $\beta=0.650274607$. After checking with a Beta Distribution PDF grapher I found online, it appears that $P(X>0.5)=0.51239$ with these particular values.

So far I've not been able to get anywhere close to this value with my numerical integration methods. I've been using the PDF of the Beta distribution to calculate the probability as follows, which I'm starting to suspect is incorrect:$$P(X>0.5)=\int_{0.5}^{1}x^{-0.323183572}(1-x)^{-0.349725393}dx$$

Using the Trapezoidal Rule with $h=0.1$ I obtained $P(X>0.5)\approx0.83334$ and with Simpson's Rule with $h= 0.05$ I obtained $P(X>0.5)\approx0.922235$. Clearly there's something wrong with my whole approach but I'm not exactly sure what so any help would be appreciated.

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For the estimated parameters you specified, it is in fact true that $\Pr[X > 0.5] = 0.51239$. What you are not doing in your calculation is multiplying by the appropriate scaling factor for the kernel of the density. In other words, the full PDF is $$f_X(x) = \color{red}{\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}} x^{\alpha - 1} (1-x)^{\beta - 1}.$$ You've forgotten the part in red.

If you do not have a built-in function to compute this scaling factor, it is simply $$\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} = \left( \int_{x=0}^1 x^{\alpha-1} (1-x)^{\beta-1} \, dx \right)^{\! -1}.$$

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