probabilitylect3

Sample Mean and Law of Large Numbers

Consider a ?nite number of random variable X, Y , . . ., Z on a sample space S. They are said to be independent

if, for any values x_i, y_j, . . . , z_k,

P (X = x_i, Y = y_j, . . . , Z = z_k) ? P (X = x_i)P (Y                 = y_j) . . . P (Z = z_k)

In particular, X and Y are independent if

P (X = x_i, Y = y_j) ? P (X = x_i)P (Y = y_j)

Now let   X  be a random variable with mean   µ. We can consider the numerical outcome of each of       n

independent trials to be a random variable with the same distribution as X. The random variable corresponding

to the ith outcome will be denoted by X_i(i    = 1, 2, . . . , n). (We note that the X_iare independent with the same

distribution as X.) The average value of all n outcomes is also a random variable which is denoted by X_nand

called the sample mean. That is:

X₁+ X₂+ · · · + X_n