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.
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
What is the formula for the probability density function?
Probability density function:In mathematics, the probability density function of a continuous random variable (which can be shortened to density function without confusion) is a function that describes the likelihood that the output value of the random variable will be in the vicinity of a definite value point.
Where in >0 is a parameter of the distribution, often referred to as the rate parameter (rateparameter). That is, the number of times an event occurs per unit of time. The interval of the exponential distribution is [o,oo). If a random variable X is exponentially distributed, it can be written: x ~ Exponential (into).
In probability theory and statistics, the exponentialdistribution is a continuous probability distribution. The exponential distribution can be used to represent the time intervals at which independent random events occur, such as the time intervals at which travelers enter an airport, the time intervals at which new entries appear on Chinese Wikipedia, and so on.
The life distributions of many electronic products generally follow an exponential distribution. The life distribution of some systems can also be approximated by the exponential distribution. It is one of the most commonly used forms of distribution in reliability studies. Exponential distribution is a special case of gamma and Weibull distribution, the failure of the product is accidental failure, its life obeys the exponential distribution.
Calculation of Probability Density Function and Distribution Function
Solution: the distribution function we generally do according to the definition: f(x) = P(X<=x);
The probability density function is derived from the distribution function: f(x) = F'(x).