Example in the title: Because Y=2X+8, Y is a monotone function about X, so we can calculate X, so X=(Y-8)/2. So you can bring X into FX(x)=FX((y-8)/2)=FY(y).
To find the probability density, you only need to take the derivative of the distribution function to get the probability density, and Fy(y) takes the derivative of y:
Fy (y) = FX ((y-8)/2) * [(derivative of y-8)/2)]
=fx((y-8)/2)*(-4), the last one is fy(y)=fx((y-8)/2)*(-4), and because fy(y)=fx((y-8)/2), the two sides are deduced separately, and finally FY (.
Extended data:
Properties of probability density:
Non-negative:
Normative:
These two basic properties can be used to judge whether a function is a probability density function of continuous random variables.
Probability refers to the probability of random occurrence of events. For uniform distribution function, the probability density is equal to the probability of an interval (the range of events) divided by the length of the interval, and its value is non-negative and can be large or small.
Properties of distribution function:
Nondegeneracy
1 and F(x) are irreducible functions. For any real number:
2. Limitations