(i) If p is uniform on (0, 1), show that 2 log(p) has the Chi-squared distribution with...

(i) If pˆ is uniform on (0, 1), show that −2 log(pˆ) has the Chi-squared distribution with 2 degrees of freedom.

(ii) Suppose pˆ1,..., pˆs are i.i.d. uniform on (0, 1). Let F = −2 log(pˆ1 ··· ˆps). Argue that F has the Chi-squared distribution with 2s degrees of freedom. What can you say about F if the pˆi are independent and satisfy P{ ˆpi ≤ u} ≤ u for all 0 ≤ u ≤ 1?

[Fisher (1934a) proposed F as a means of combining p-values from independent experiments.]