Background A quick calculation yields, E[Mn]=u,var(Mn)= 02 The variance of M decreases to zero as n increases. Thus the bulk of the distribution of M must be very close to the mean u. This phenomenon is the subject of certain laws of large numbers
Background A quick calculation yields, 𝐄[𝑀𝑛] = 𝜇, var(𝑀𝑛) = 𝜎 2 𝑛 . The variance of 𝑀𝑛 decreases to zero as 𝑛 increases. Thus the bulk of the distribution of 𝑀𝑛 must be very close to the mean 𝜇. This phenomenon is the subject of certain laws of large numbers
Background The laws generally assert that the sample mean Mn converges to the true mean u. These laws provide a mathematical basis for the loose interpretation of an expectation E X=u... .. as the average of a large number of independent samples drawn from the distribution of X
Background The laws generally assert that the sample mean 𝑀𝑛 converges to the true mean 𝜇. These laws provide a mathematical basis for the loose interpretation of an expectation 𝐄[𝑋] = 𝜇 … … as the average of a large number of independent samples drawn from the distribution of 𝑋
Background We will also consider a quantity which is intermediate between Sn and Mn. Zm is defined as follows. 1. subtract nu from S,to obtain the zero-mean random variable Sm-nu 2.then divide by ovn,to form the random variable Sn -nu In= 0√m
Background We will also consider a quantity which is intermediate between 𝑆𝑛 and 𝑀𝑛. 𝑍𝑛 is defined as follows. 1. subtract 𝑛𝜇 from 𝑆𝑛, to obtain the zero-mean random variable 𝑆𝑛 − 𝑛𝜇 2. then divide by 𝜎 𝑛, to form the random variable 𝑍𝑛 = 𝑆𝑛 − 𝑛𝜇 𝜎 𝑛
Background -It can be seen that E[Zn]=0, var[Zn]=1 ■ Since the mean/variance of Zn remain unchanged as n increases,its distribution neither spreads,nor shrinks to a point The central limit theorem is concerned with 口 the asymptotic shape of the distribution of Zm and asserts that Z,becomes the standard normal distribution
Background It can be seen that 𝐄 𝑍𝑛 = 0, var 𝑍𝑛 = 1 Since the mean/variance of 𝑍𝑛 remain unchanged as 𝑛 increases, its distribution neither spreads, nor shrinks to a point. The central limit theorem is concerned with the asymptotic shape of the distribution of 𝑍𝑛 and asserts that 𝑍𝑛 becomes the standard normal distribution
Application Limit theorems are useful for several reasons: ■ (a)Conceptually.They provided an interpretation of expectations/probabilities in terms of a long sequence of identical independent experiments
Application Limit theorems are useful for several reasons: (a) Conceptually. They provided an interpretation of expectations/probabilities in terms of a long sequence of identical independent experiments