Suppose that there are multiple potential candidates for the PDF of PX : pj
j∈[m], where pj : R →R≥0 has the property RR pj (x) ML (dx) = 1, for each j ∈ [m].(d) Construct a dierent rejection rule of size α for testing the same hypothesis H0 as in Part(c), but which instead makes comparisons between p0 and all alternative candidates:pj
j∈[m]. HintUse the fact that E (X + Y ) = E (X)+E (Y ), for any random variablesX and Y .

The question seems to be about constructing a rejection rule for hypothesis testing, where the null hypothesis H0 is that the probability density function (PDF) of a random variable X is p0, and the alternative hypotheses are that the PDF of X is one of the other pj, j ∈ [m]. The rejection rule should have a size α, meaning that the probability of rejecting H0 when it is true is at most α.

Here are the steps to construct such a rejection rule:

1. Define the test statistic: The test statistic should be a function of the observed data that helps us decide between the null and alternative hypotheses. In this case, a natural choice for the test statistic might be the likelihood ratio L = p0(X) / maxj pj(X), where X is the observed data.

2. Define the rejection region: The rejection region is the set of values of the test statistic for which we reject H0. Since we want to reject H0 when the likelihood of the data under p0 is much smaller than under the best alternative, we could define the rejection region as {L < c}, for some constant c.

3. Choose the constant c: The constant c should be chosen such that the size of the test is α. This means that the probability of rejecting H0 when it is true (i.e., when the PDF of X is p0) should be at most α. This can be written as P(L < c | H0) ≤ α. To find such a c, we can use the cumulative distribution function (CDF) of L under H0, denoted by F0. We need to find c such that F0(c) = α.

4. Apply the test: Once the test statistic, the rejection region, and the constant c have been defined, the test can be applied to any observed data. If the test statistic falls in the rejection region, we reject H0; otherwise, we do not reject H0.

Note: The hint about the expectation of the sum of two random variables seems to be unrelated to the main question. It might be relevant to a different part of the problem.