Transformation of random variables vs shift of functions

1. What is the transformation of random variables?
2. What is the effect on a random variable of adding or subtracting by a constant?
3. What is variable transformation?

What is the transformation of random variables?

The transformation y=a+Bx maps Rn one-to-one and onto Rn. The inverse transformation is x=B−1(y−a). The Jacobian of the inverse transformation is the constant function det(B−1)=1/det(B). The result now follows from the multivariate change of variables theorem.

What is the effect on a random variable of adding or subtracting by a constant?

When adding or subtracting a constant to/from a random variable, the mean is changed directly by the constant, but the standard deviation remains unchanged. Since we are simply adding a constant, 3.5, to , the standard deviation remains unchanged, so σ X + 3.5 = 2.4 .

What is variable transformation?

In data analysis transformation is the replacement of a variable by a function of that variable: for example, replacing a variable x by the square root of x or the logarithm of x. In a stronger sense, a transformation is a replacement that changes the shape of a distribution or relationship.