Several important probability distributions (lower)

The previous article introduced two distributions and Poisson distributions.

This article will introduceUniform Distribution,Exponential DistributionAndNormal Distribution.

3. Uniform Distribution (Uniform)

If the random variable XDensity functionsIs

The random variable X follows an even distribution in the range [a, B. As x ~ U (A, B ).

Shows the image:

 

 

 

 

 

 

 

 

 

 

 

 

 

Evenly distributedDistribution FunctionsIs

Shows the image:

 

 

 

 

 

 

 

 

 

 

 

 

 

Mathematical expectation of uniform distribution E (x) = 1/(2 * (B + a), variance is d (x) = 1/(12 * (B-) 2 ).

4. Exponential Distribution

If the random variable X'sDensity functionsIs

Where λ> 0 is a constant, the random variable X follows the exponential distribution of the parameter λ. Shows the density function image:

 

 

 

 

 

 

 

 

 

 

 

 

 

Exponential DistributionDistribution FunctionsIs:

 

Mathematical expectation E (x) = 1/λ, variance is d (x) = 1/λ 2. Shows the histogram image of exponential distribution:

 

 

 

 

 

 

 

 

 

 

 

 

 

The larger the value of λ, the faster the slope of the curve changes.

 

5. Normal Distribution

If the density function of the continuous random variable X is

Here,-∞ <x <+ ∞, and-∞ <μ <+ ∞, σ are parameters. The random variable X follows the normal distribution of the parameter (μ, σ 2) and is recorded as X ~ N (μ, σ 2)

If μ = 0, σ = 1, n (0, 1) is the standard normal distribution.

Normal Distribution has the following features:

① When μ changes but σ remains unchanged, the image moves along the X axis, and the image shape does not change.

 

 

 

 

 

 

 

 

 

 

 

 

 

② μs remain unchanged, while σ changes, the position of the image remains unchanged, but the morphology changes. The larger σ, the fatter the image.

 

 

 

 

 

 

 

 

 

 

 

 

 

③ Curves have inflection points at points x = μ-σ and x = μ + σ.