Relationship between probability density function and probability function

What is the relationship between probability function and probability density and distribution function?

Set: The probability distribution function is: F(x)

The probability density function is: f(x)

The relationship between the two is:

f(x)=dF(x)/dx

That is, the density function, f, is the first-order derivative of the distribution function F. Or the distribution function is the integral of the density function.

Probability function and probability density and distribution function in the end what is the relationship between the distribution function, for concise answer

Distribution function is defined as follows:

Definition of the function F(x)=P{X<=x}(Note: is less than or equal to, ensure that the right of the F(x) continuous).

Then such as for the distribution function F(x) of the random variable X, if there exists a non-negative function f(x).

So that for any real number x, there is F (x) = ∫ (-∞, x) f (t) dt then X becomes a continuous random variable.

Where the function f(x) is called the probability density function of X, or probability density for short. This is the definition of probability density.

Example:

It is known that the two-dimensional random variable (X,Y) has a probability density f(x,y)=2e-(2x+y),x>0,y>0

0, others

Seek the joint distribution function F(x,y) marginal probability density fx(x) and fy(y)

Determine whether X in Y are mutually are independent of each other.

Solution:

F(x, y)

=2∫(0,x)e^(-2x)dx∫(0,y)e^(-y)dy

=(e^(-2x)-1)*(e^(-y)-1)*(e^(-2x)-1)*(e^(-y)-1)*(e^(-y)-1). ^(-y)-1)

fx(x)

=2∫(0,∞)e^(-2x)e^(-y)dy

=2e^(-2x)

fy(y)

=2∫(0,∞)e^(-2x)e^(-y)dx

=e^(-y)

X and Y are independent of each other.

Extended Information

Difference between Probability Density and Probability Density Function:

Probability refers to the chances of an event occurring randomly, and the concept of probability density is roughly the same, referring to the probability distribution of an event.

In mathematics, the probability density function (which can be shortened to density function when it is not so confusing) of a continuous random variable is a function that describes the likelihood that the output value of this random variable, will be in the vicinity of some definite point of value. probabilitydensityfunction, or PDF for short.

Probability density functions add up to probability function (discrete variable), or integral (continuous variable).

In mathematics, the probability density function (which can be abbreviated to density function when not confusing) of a continuous random variable is an output value that describes that random variable.

A function that describes the likelihood that it will fall in the vicinity of some definite point of value. And the probability that the value of the random variable falls within a certain region is the integral of the probability density function over that region.

When a probability density function exists, the cumulative distribution function is the integral of the probability density function. The probability density function is usually labeled in lower case.

Definition:

For a one-dimensional real random variable X, let its cumulative distribution function be, If there exists a measurable function that satisfies:, then X is a continuous random variable and is its probability density function.

What is the relationship between probability density and distribution function?

The graph of a probability density function is “bounded” (if it is not bounded, it is not cumulative, i.e., its distribution would not exist), while the graph of a distribution function is unbounded.

Mathematically, the distribution function F(x)=P(X<=x)

The probability density f(x) is the first-order derivative of F(x) with respect to x at x, i.e. the rate of change. If you take a very small neighborhood Δx around a certain x, then the probability that the random variable X falls within (x,x+Δx) is approximately f(x)Δx, i.e., P(x<X<x+Δx)

In other words, the probability density f(x) is the probability that X will fall within a “unit width” at x. The term “density” can be understood in this way.