Uniform probability distribution. Converting a uniformly distributed random variable to a normally distributed one
A distribution is considered uniform when all values of a random variable (in the area of its existence, for example, in an interval) are equally probable. The distribution function for such a random variable is:
Distribution density: 
1
Rice. Plots of the distribution function (left) and distribution density (right).
Uniform distribution - concept and types. Classification and features of the category "Equal distribution" 2017, 2018.
Basic discrete distributions of random variables Definition 1. A random variable X, taking values 1, 2, ..., n, has even distribution if Pm = P (X = m) = 1 / n, m = 1,…, n. It's obvious that. Consider the following problem: an urn contains N balls, of which M are white balls ....
Distribution laws of continuous random variables Definition 5. Continuous random value X, which takes a value on a segment, has a uniform distribution if the distribution density has the form. (1) It is easy to see that,. If a random variable ....
A distribution is considered uniform when all values of a random variable (in the region of its existence, for example, in an interval) are equally probable. The distribution function for such a random variable has the form: Distribution density: F (x) f (x) 1 0 a b x 0 a b x ....
Normal distribution laws Uniform, exponential and The probability density function of the uniform law is as follows: (10.17) where a and b are given numbers, a< b; a и b – это параметры равномерного закона. Найдем функцию распределения F(x)... .
Uniform probability distribution is the simplest and can be either discrete or continuous. Discrete uniform distribution is a distribution for which the probability of each of the values of SV is the same, that is: where N is the number ....
Definition 16: A continuous random variable has a uniform distribution on an interval if the distribution density of a given random variable is constant on this interval, and outside it is equal to zero, that is, (45) The density graph for a uniform distribution is shown ...
Distribution of probabilities of a continuous random variable X taking all values from the segment is called uniform if its probability density on this interval is constant, and outside it is equal to zero. Thus, the probability density of a continuous random variable X uniformly distributed on the segment , has the form:
We define expected value , variance and for a random variable with a uniform distribution.
, 
, 
.
Example. All values of a uniformly distributed random variable lie on the segment ... Find the probability of a random variable hitting the interval (3;5) .
a = 2, b = 8, 
.
Binomial distribution
Let it be produced n tests, and the probability of occurrence of an event A in each trial is p and does not depend on the outcome of other tests (independent tests). Since the probability of an event occurring A in one trial is p, then the probability of its non-occurrence is q = 1-p.
Let the event A came in n trials m once. This complex event can be written as a work:
.
Then the likelihood of that at n test event A will come m times, is calculated by the formula:
or 
(1)
Formula (1) is called by the Bernoulli formula.
Let be X- random value, equal to the number event appearances A v n tests, which takes values with probabilities:
The resulting distribution law for a random variable is called binomial distribution law.
| X | m | n | ||||
| P |
Expected value, dispersion and the average standard deviation random variables distributed according to the binomial law are determined by the formulas:
, , .
Example. Three shots are fired at the target, and the probability of hitting each shot is 0.8. Consider a random variable X- the number of hits on the target. Find its distribution law, mathematical expectation, variance and standard deviation.
p = 0.8, q = 0.2, n = 3, 
, , 
.
- probability of 0 hits;
The probability of one hit;
The probability of two hits;
- the probability of three hits.
We get the distribution law:
| X | ||||
| P | 0,008 | 0,096 | 0,384 | 0,512 |
Tasks
1. The coin is thrown 7 times. Find the probability that it will fall with the coat of arms up 4 times.
2. The coin is thrown 8 times. Find the probability that the coat of arms will be drawn no more than three times.
3. The probability of hitting the target when firing from a gun p = 0.6. Find the mathematical expectation of the total number of hits if 10 shots are fired.
4. Find the mathematical expectation of the number of lottery tickets that will win if 20 tickets are purchased, and the probability of winning for one ticket is 0.3.
As an example of a continuous random variable, consider a random variable X uniformly distributed over the interval (a; b). The random variable X is said to be evenly distributed on the interval (a; b), if its distribution density is not constant on this interval:
From the normalization condition, we determine the value of the constant c. The area under the distribution density curve should be equal to one, but in our case it is the area of a rectangle with a base (b - α) and a height c (Fig. 1). 
Rice. 1 Density of uniform distribution
From here we find the value of the constant c: 
So, the density of a uniformly distributed random variable is 
Let us now find the distribution function by the formula:
1) for 
2) for
3) for 0 + 1 + 0 = 1.
Thus, 
The distribution function is continuous and does not decrease (Fig. 2). 
Rice. 2 Distribution function of a uniformly distributed random variable
Find mathematical expectation of a uniformly distributed random variable according to the formula:
Dispersion of uniform distribution is calculated by the formula and is equal to
Example # 1. Scale division measuring instrument is equal to 0.2. Instrument readings are rounded to the nearest whole division. Find the probability that an error will be made during the counting: a) less than 0.04; b) large 0.02
Solution. Round-off error is a random variable evenly distributed over the interval between adjacent integer divisions. Let us consider the interval (0; 0.2) as such a division (Fig. A). Rounding can be carried out both towards the left border - 0, and towards the right - 0.2, which means that an error less than or equal to 0.04 can be made twice, which must be taken into account when calculating the probability: 
P = 0.2 + 0.2 = 0.4
For the second case, the magnitude of the error can also exceed 0.02 on both division boundaries, that is, it can be either more than 0.02 or less than 0.18. 
Then the probability of such an error:
Example # 2. It was assumed that the stability of the economic situation in the country (the absence of wars, natural disasters, etc.) over the past 50 years can be judged by the nature of the distribution of the population by age: in a calm environment, it should be uniform... As a result of the study, the following data were obtained for one of the countries.
Are there any grounds to believe that there was an unstable situation in the country?The solution is carried out using the calculator Hypothesis testing... Table for calculating indicators.
| Groups | Midpoint of the interval, x i | Quantity, f i | x i * f i | Accumulated frequency, S | | x - x cf | * f | (x - x cf) 2 * f | Frequency, f i / n |
| 0 - 10 | 5 | 0.14 | 0.7 | 0.14 | 5.32 | 202.16 | 0.14 |
| 10 - 20 | 15 | 0.09 | 1.35 | 0.23 | 2.52 | 70.56 | 0.09 |
| 20 - 30 | 25 | 0.1 | 2.5 | 0.33 | 1.8 | 32.4 | 0.1 |
| 30 - 40 | 35 | 0.08 | 2.8 | 0.41 | 0.64 | 5.12 | 0.08 |
| 40 - 50 | 45 | 0.16 | 7.2 | 0.57 | 0.32 | 0.64 | 0.16 |
| 50 - 60 | 55 | 0.13 | 7.15 | 0.7 | 1.56 | 18.72 | 0.13 |
| 60 - 70 | 65 | 0.12 | 7.8 | 0.82 | 2.64 | 58.08 | 0.12 |
| 70 - 80 | 75 | 0.18 | 13.5 | 1 | 5.76 | 184.32 | 0.18 |
| 1 | 43 | 20.56 | 572 | 1 |
Weighted average
Variation indicators.
Absolute indicators of variation.
The range of variation is the difference between the maximum and minimum values of the primary series feature.
R = X max - X min
R = 70 - 0 = 70
Dispersion- characterizes the measure of dispersion around its mean value (the measure of dispersion, i.e. deviation from the mean).
Standard deviation.
Each value of the series differs from the mean value 43 by no more than 23.92
Testing hypotheses about the type of distribution.
4. Testing the hypothesis about even distribution general population.
In order to test the hypothesis about the uniform distribution of X, i.e. according to the law: f (x) = 1 / (b-a) in the interval (a, b)
necessary:
1. Estimate the parameters a and b - the ends of the interval in which the possible values of X were observed, according to the formulas (the parameter estimates are denoted through the * sign):
2. Find the probability density of the assumed distribution f (x) = 1 / (b * - a *)
3. Find theoretical frequencies:
n 1 = nP 1 = n = n * 1 / (b * - a *) * (x 1 - a *)
n 2 = n 3 = ... = n s-1 = n * 1 / (b * - a *) * (x i - x i-1)
n s = n * 1 / (b * - a *) * (b * - x s-1)
4. Compare empirical and theoretical frequencies using Pearson's criterion, taking the number of degrees of freedom k = s-3, where s is the number of initial sampling intervals; if the combination of few frequencies was made, and hence the intervals themselves, then s is the number of intervals remaining after the combination.
Solution:
1. Let us find the estimates of the parameters a * and b * of the uniform distribution by the formulas:
2. Find the density of the assumed uniform distribution:
f (x) = 1 / (b * - a *) = 1 / (84.42 - 1.58) = 0.0121
3. Let's find the theoretical frequencies:
n 1 = n * f (x) (x 1 - a *) = 1 * 0.0121 (10-1.58) = 0.1
n 8 = n * f (x) (b * - x 7) = 1 * 0.0121 (84.42-70) = 0.17
The rest n s will be equal:
n s = n * f (x) (x i - x i-1)
| i | n i | n * i | n i - n * i | (n i - n * i) 2 | (n i - n * i) 2 / n * i |
| 1 | 0.14 | 0.1 | 0.0383 | 0.00147 | 0.0144 |
| 2 | 0.09 | 0.12 | -0.0307 | 0.000943 | 0.00781 |
| 3 | 0.1 | 0.12 | -0.0207 | 0.000429 | 0.00355 |
| 4 | 0.08 | 0.12 | -0.0407 | 0.00166 | 0.0137 |
| 5 | 0.16 | 0.12 | 0.0393 | 0.00154 | 0.0128 |
| 6 | 0.13 | 0.12 | 0.0093 | 8.6E-5 | 0.000716 |
| 7 | 0.12 | 0.12 | -0.000701 | 0 | 4.0E-6 |
| 8 | 0.18 | 0.17 | 0.00589 | 3.5E-5 | 0.000199 |
| Total | 1 | 0.0532 |
Therefore, the critical area for this statistic is always right-handed :)
