Question

4. Consider the sample space S 1,2,3,...), and assume that outcomes have the probabilities P(i)- 2-'. For any n 2 0, define the discrete random variable Xn S0,... , n) by x,(i)-1 mod (n + 1), where mod means"modulo (a) Show that Xn converges in probability to the "identity" random variable X, defined by X(i)-. (b) Show that Xn converges in distribution to the Geom (1/2) random variable (e.g. to the time of the first Head in a sequence of fair coin flips).

Answer 1

Two ways to increase the statistical power of an experiment overlap in the two sampling distributions for null hypothesis and alternative hypothesis Larger the sample size, the higher the power 2. Decrease random error - Reduce the variance: smaller variance yields higher power. Power is higher when the standard deviation is smaller than when it applied consistently.