Could you help giving me explanations on this problem ? Thank you.

Let X be an unknown random variable. We ran an experiments for X , we denote bv Y; (2' 2 D) the number of samples that are equal to i. (a). Suppose that X is a binomial random variable M4, p) with p unknown. Show that the maximum likelihood estimate if is equal to A Y1+2Y2+3Y3+4Kl p: 4?; (Give a rigorous proof.) b) Show that the formula for p in part (a) is a sufficient statistic for p. (Give a rigorous proof.)(c) We took n = 32 samples of X, and we get Yo = 2; Y1 = 8; Y2 = 8; Y3 = 8; Y4 = 6. Suppose that the hypotheses are . Null hypothesis Ho: X is a binomial random variable b(4, p) for some unknown p. . Alternative hypothesis H1: the null hypothesis is false. Can we reject the null hypothesis at a = 0.05 significance level