(c) Explain the Dominated Convergence Theorem (DCT) without providing a proof.

(b) Define the Integral of a non-negative random variable and state the associated well-known theorem.

27. Prove that if Y, ≥0, then E (Σκ) - Σει

25. (a) Define the Integral of a simple random variable and provide an illustrative example.

(b) Explain the Monotone Convergence Theorem (MCT) without providing a proof.

28. (a) Under what conditions do we say that two random variables X and Y are independent?

The masses measured on a population of 100 animals were grouped in the following table, after being recorded to the nearest gram Mass 89 90-109 110-129 130-149 150-169 170-189 > 190 Frequency 3 7 34 43 10 2 1 You are given that the sample mean of the data is 131.5 and the sample standard deviation is 20.0. Test the hypothesis that the distribution of masses follows a normal distribution at the 5% significance level.

state without proof the uniqueness theorm of probability function

(a+b) R2L 2+2*0=? Ma state without proof the uniqueness theorm of probability function suppose thatPandQ are probability measures defined on the same probability space (Q, F)and that Fis generated by a π-system if P(A)=Q(A) tax for all A EthenP=Q i. e. P(A)=Q(A) for alla g // معدلة 2:23 ص