Question

Let : be a random variable that follows a distribution to be specified later Suppose that , 2 are n independent, and identically distributed (i.i.d.) realizations of z. In what follows, a, b, c, and dare constants. (1) (12 pts) Suppose that: follows a Bernoulli distribution, i.e., P(z-1)-p and a. Write El-l and var() as functions of p. b. Suppose, among (), there are n samples with 1 and no samples with 0. Write down the sample mean and the sample variance as functions of ni and no . State the law of large numbers and the central limit theorem for the sample mean (note: write any version you learned from previous courses).

Answer 1

c) The law of large numbers states that, as the number of I.I.D. random variables increases, their sample mean (average) approaches their theoretical mean.

Central limit theorem states that, for sufficiently large samples from a population with finite variance, the mean of all samples from the same population will be approximately equal to the mean of the population.