# How to find the probability density function by the formula method

### What is the formula for probability density?

Probability density: f(x)=(1/2√π)exp{-(x-3)²/2*2}

Based on the expression for the normal probability density function in the question you can immediately get the mathematical expectation and the variance of the random variable:

Mathematical Expectation: μ=3

Variance: σ²=2

The continuous type random variable’s The probability density function (which can be shortened to density function when it is not so confusing) 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.

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 lowercase.

Extended information:

The probability density function of random data represents the probability that the instantaneous amplitude falls within a specified range. It is therefore a function of the magnitude. It varies with the amplitude of the range taken.

The probability density function f(x) has the following properties:

(1)f(x)≧0;

(2)∫f(x)d(x)=1;

(3)P(a<X≦b)=∫f(x)dx.

### 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.

Formula:

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).

Distribution:

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.

### Probability density function formula?

E(X)=X1*p(X1)+X2*p(X2)+……+Xn*p(Xn)=X1*f1(X1)+X2*f2(X2)+……+Xn*fn(Xn).

X; 1,X; 2,X; 3, ……, X.

n is this discrete random variable, p(X1),p(X2),p(X3), ……p(Xn) is the probability function of these several data. In the random occurrence of several data p(X1),p(X2),p(X3),……p(Xn) probability function is understood as the frequency f(Xn) of the occurrence of data X1,X2,X3,……,Xn.

Variance of common distributions

1. Two-point distribution.

2. Binomial distribution X~B(n,p) introduces the random variable Xi (the number of times A occurs in the ith trial, obeying a two-point distribution).

3, Poisson distribution (derivation omitted).

4, uniform distribution Another calculation procedure is.

5, exponential distribution (derivation omitted).

6, normal distribution (derivation omitted).

7, t distribution: where X ~ T (n), E (X) = 0.

8, F distribution: where X ~ F (m,n).