How do you do the definite integral of a probability density function?
Where is your probability density function? In the topic of probability theory, remember the basic concept of integrating the probability density function, what you get is the distribution function i.e. ∫ (negative infinity to x) f(x)dx=F(x) and the expected value E(x)=∫ (negative infinity to positive infinity) x*f(x)dx
If the probability density function is f(x) and F'(x)=f(x), then the probability distribution function is F(x)+C, C is a constant, can be based on x tends to infinity when the probability distribution function is equal to 1 to find
Answers to the steps have been relatively detailed, the probability density for the definite integral to get the distribution function.
After substituting the formulas, both of those answers are directly derived using the basic calculation of the definite integral.
How to find the probability density 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.
The distribution function is defined because in many cases we do not want to know the probability of something being at a particular value, but at best we want to know the probability of it being in a certain range, and so the concept of the distribution function is introduced.
And the probability density, if continuous at x. It is the distribution function F(x) that is derived from x. Conversely, knowing the probability density function, the distribution function can be derived by integrating negative infinity to x.
Probability density:
There is no practical significance in speaking simply of the probability density, which must be predicated on a definite bounded interval. You can think of the probability density as the vertical coordinate, the interval as the horizontal coordinate, the integral of the probability density to the interval is the area, and this area is the probability of the event occurring in this interval, the sum of all the areas is 1. So analyzing the probability density of a point alone does not have any significance, it has to have the interval as a reference and comparison.
Reference to the above: Baidu Encyclopedia – Probability Density
How do you find the integral of the probability density of a normal distribution?
The probability density function of a normal distribution is the integral value of f(x) from negative infinity to positive infinity.1.
Simply make the equation normal distribution with mean μ=0 and standard deviation σ=1/root sign 2.Then that normal-too-distributed probability density function becomes f(x)=(1/root sign π)*e^(-x^2) it integrates from negative infinity to positive infinity as the value of 1.
< p>Therefore, the required integral: e^(-x^2) which integrates from negative infinity to positive infinity has an integral value of the root sign π.
Expansion
The integral of a function expresses the overall properties of the function over a region, and changing the value of the function at a given point does not change its integral value. For a Riemann productable function, changing the value of a finite number of points leaves its integral unchanged. For a Leberger productable function, a change in the value of the function on a set of measure 0 does not affect the value of its integral.
If two functions are the same almost everywhere, then they have the same integral. If, for any element A in f, the integral of the productable function f over A is always equal to (greater than or equal to) the integral of the productable function g over A, then f is equal to (greater than or equal to) g almost everywhere.