The Function of a Random Variable

F(X) is a transformed version of X. You should not assume that F(X) will have the same probability distribution function than X.

In other words, the mapping or transform of X, leads to another probability distribution function. Keep in mind these two important

ideas:

• if X is a random variable, any function of X, F(X), will also be random.

• X and F(X) have unique(各自) probability distribution (unless F(X) = X of course).

（F(X) 是 随机变量X的一个函数，但是F(X) 的概率分布 和 X的概率分布 并不一样的，有各自的概率分布）

Example:

we know that in the case of a uniformly distributed random variable with possible outcome {1, 2, 3, 4, 5, 6}, the probability of each outcome is 1/6. If the function F(X) is defined as (X - 3) ^ 2, Let's compute the probability distribution of F(X).

F(X) = (X - 3)^2

As you can see in the table above, computing F(X) for each outcome in X, results in one zero, two ones, two fours and one nine. The probability of an outcome Y from F(X) is equal to the sum of the probability of any of the X for which F(X) = Y. Thus we get:

Now, if we wish to compute the expected (期望值)value of F(X) we can proceed(开始) in two ways. If we know the probability distribution of F(X) we can write (method 1):

（上面是 先 利用 随机变量 X 来计算 F(X) 的 概率分布，从而计算F(X)的数学期望）

Mathematically, if we call Y the random variable F(X)

Or if you don't know the probability distribution of F(X), you can still use your knowledge of the probability distribution of X to

calculate E[Y] (method 2):

Mathematically we would write this result as:

This in an important result because, in practice, you don't necessarily know the probability distribution of F(X). Of

course you can calculate it, but this is an extra(额外) step, which you can avoid if you use the second method.

Mathematically, this result can be written as:

for a discrete random variable

for continuous random variables

Keep in mind that the beauty of the method is that it is not necessary to know the probability distribution of F(X) to

compute its expected value, as long as you know the probability distribution of X.

（上面计算 F(X)的数学期望，是使用X的概率分布，不需要去计算 F(X)的概率分布）