Use Theorem 6.7 to draw a random sample of n = 5 observations from a beta distribution with α = 2 and β = 1.

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THEOREM 6.7 Let Y be a continuous random variable with density function f( y) and cumulative dis- tribution F(y). Then the density function of W = F(y) will be a uniform distribution defined over the interval 0 < w < 1, i.e., g(w) = 1 (0 sws 1) Proof of Theorem 6.7 Figure 6.10 shows the graph of W = F(y) for a continuous random variable Y. You can see from the figure that there is a one-to-one correspon- dence between y values and w values, and that values of Y corresponding to values of W in the interval 0 < Wsw will be those in the interval 0 < Y< y. Therefore, P(W s w) = P(Y s y) = F(y) %3| But since W = F(y), we have F(y) = w. Therefore, we can write G(w) = P(W < w) = F(y) = w Finally, we differentiate over the range 0