The probability of a random variable falling into an interval. Probability of a normally distributed random variable falling into a given interval

It is already known that if a random variable X is given by the distribution density f (x), then the probability that X will take a value belonging to the interval (a, b) is as follows:

Let the random variable X be distributed according to the normal law. Then the probability that X will take a value belonging to the interval (a,b) is equal to

Let's transform this formula so that you can use ready-made tables. Let's introduce a new variable z = (x--а)/--s. Hence x = sz+a, dx = sdz. Let's find new limits of integration. If x= a, then z=(a-a)/--s; if x = b, then z = (b-a)/--s.

Thus we have

Using the Laplace function

we'll finally get it

Calculating the probability of a random event

In a batch of 14 parts there are 2 non-standard parts. 3 items were selected at random. Draw up a distribution law for the random variable X - the number of standard parts among the selected ones. Find numerical characteristics, . The solution is obvious...

Research on the tensile strength of calico strips

They say...

Methods for estimating unknown distribution parameters

If a random variable X is given by a distribution density, then the probability that X will take a value belonging to the interval is as follows: Let the random variable X be normally distributed. Then the probability that X will take the value...

Continuous random variable

The probability distribution function F(x) of a random variable X at point x is the probability that, as a result of an experiment, the random variable will take on a value less than x, i.e. F(x)=P(X< х}. Рассмотрим свойства функции F(x). 1. F(-?)=lim(x>-?)F(x)=0...

Continuous random variables. Normal distribution law

Knowing the distribution density, you can calculate the probability that a continuous random variable will take a value belonging to a given interval. The calculation is based on the following theorem. Theorem. The likelihood...

Final mathematical expectation mx=5 Standard deviation yx=3 Sample size n=335 Confidence probability r=0.95 Significance level Number of selected values ​​N=13 Modeling a random variable...

Static system modeling

Static system modeling

3. Estimation of statistical characteristics of a random process. Problems are determined according to sections...

Static system modeling

Distribution: f(x)=b(3-x), b>0 Distribution boundaries 1