A discrete-valued random variable X takes values in 0, 1, 2, . . . , where p(X = i) = πi.

(a) Write down formulas for:

• the p-value at X = i
• the probability distribution function (pdf) of p-values
• the cumulative distribution function (cdf) of p-values.

(b) Calculate and plot the cdf of p-values for the Poisson distribution with parameter λ = 3.5.

(c) Repeat part (b) for λ= 5.0.

(d) Calculate and plot the cdf of p-values for a 2-component

Poisson mixture model with parameters λ1 = 3.5, λ2 = 5.0 and component weights δ1 = 0.4, δ2 = 0.6.