Probability/Transformation of Random Variables

Underlying principle
Let $$X_1,\dotsc,X_n$$ be $$n$$ random variables, $$Y_1,\dotsc,Y_n$$ be another $$n$$ random variables, and $$\mathbf X=(X_1,\dotsc,X_n)^T,\mathbf Y=(Y_1,\dotsc,Y_n)^T$$ be random (column) vectors.

Suppose the vector-valued function $$\mathbf g:\operatorname{supp}(\mathbf X)\to\operatorname{supp}(\mathbf Y)$$ is bijective (it is also called one-to-one correspondence in this case). Then, its inverse $$\mathbf g^{-1}:\operatorname{supp}(\mathbf Y)\to\operatorname{supp}(\mathbf X)$$ exists.

After that, we can transform $$\mathbf X$$ to $$\mathbf Y$$ by applying the transformation $$\mathbf g$$, i.e. by $$\mathbf Y=\mathbf g(\mathbf X)$$, and transform $$\mathbf Y$$ to $$\mathbf X$$ by applying the inverse transformation $$\mathbf g^{-1}$$, i.e. by $$\mathbf X=\mathbf g^{-1}(\mathbf Y)$$.

We are often interested in deriving the joint probability function $$f_{\mathbf Y}(\mathbf y)$$ of $$\mathbf Y$$, given the joint probability function $$f_{\mathbf X}(\mathbf x)$$ of $$\mathbf X$$. We will examine the and  cases one by one in the following.

Transformation of continuous random variables
For random variables, the situation is more complicated.

Let us investigate the case for univariate pdf, which is simpler.

Let us define, and introduce several notations in the definition.

Joint moment generating function
In the following, we will use $$\mathbf X$$ to denote $$(X_1,\dotsc,X_n)^T$$.

Analogously, we have mgf.

Distribution of linear transformation of random variables
We will prove some propositions about distributions of linear transformation of random variables using. Some of them are mentioned in previous chapters. As we will see, proving these propositions using mgf is quite simple.

Central limit theorem
We will provide a proof to (CLT) using mgf here.

A special case of using CLT as is using normal distribution to  discrete distribution. To improve accuracy, we should ideally have, as explained in the following.

Illustration of continuity correcction: i-1/2 i i+1/2

i-1   i

i    i+1