### What’s the mathematical expectation and variance of the 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 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.

### Expectation, variance, coefficient of variance of normal probability density function

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

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

Mathematical expectation: μ=3

Variance :σ²= 2

### What is the formula for the variance of a random variable X?

The equation D(X)=E{[X-E(X)]^2}=E(X^2)-[E(X)]^2, where E(X) denotes mathematical expectation.

For a continuous random variable X, if its domain of definition is (a, b) and the probability density function is f(x), the formula for calculating the variance of a continuous random variable X is D(X)=(x-μ)^2f(x)dx.

The variance depicts the degree of dispersion of the values of a random variable with respect to its mathematical expectation. (The larger the standard deviation and variance, the greater the dispersion.) If the values of X are more concentrated, the variance D(X) is smaller, and if the values of X are more dispersed, the variance D(X) is larger. Therefore, D(X) is a quantity that portrays the degree of dispersion of the values of X. It is a measure of the degree of dispersion of the values.

Extended information:

Covariance of commonly used distributions

1, two-point distribution

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

3, Poisson distribution (the derivation of which is 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).

### How to find the mathematical expectation and variance of a known probability density function

Substitute the formula. Uniform distribution on [a,b], expectation = (a+b)/2, variance = [(b-a)^2]/2. Substitution is straightforward. If you don’t know the expectation and variance formulas for the uniform distribution, you can only do it step-by-step:

Expectation:

EX=∫{product from -a to a}xf(x)dx

=∫{product from -a to a}x/2adx

=x^2/4a|{upper a,lower -a}

=0

E(X^2 )=∫{Accumulate from -a to a}(x^2)*f(x)dx

=∫{Accumulate from -a to a}x^2/2adx

=x^3/6a|{Up-a,Down-a}

=(a^2)/3

Variance:

DX=E(X^2)-(EX)^2=(a^2)/3

Extended information:

Discrete random variables and continuous random variables are both determined by the range of values the random variable takes (takes).

Variables that take values that can only take discrete natural numbers are discrete random variables. For example, a toss of 20 coins, k coins face up, k is a random variable. k can only take the value of the natural numbers 0, 1, 2, …, 20, and can not take the decimal 3.5, irrational numbers, and therefore k is a discrete random variable.

If the variable can take any real number in a certain interval, i.e., the values of the variable can be continuous, this random variable is called a continuous random variable.

For example, if a bus runs every 15 minutes and a person waits for the bus at the platform for x, which is a random variable with values in the range [0,15), it is an interval within which any real number 3.5, irrational number, etc., can be taken, and the random variable is thus said to be a continuous random variable.

Since the value of the random variable X depends only on the integral of the probability density function, the value of the probability density function at individual points does not affect the performance of the random variable.

### Find the variance of the probability density function

First, find c=2 by the property that the probability density integrates to 1, and then find the variance of the random variable by the formula.

Refer to the figure below for the procedure and answer.