1. Continuous distributions
Practice-4
Additional chapters of mathematics
Dmitriy Sergeevich Nikitin
Assistant
Tomsk Polytechnic University

2. Normal Distribution
This is the most important continuous distribution because in applications many random variables are normal random variables (that is, they have a normal distribution) or they are approximately normal or can be transformed into normal random variables in a relatively simple fashion. Furthermore, the normal distribution is a useful approximation of more complicated distributions, and it also occurs in the proofs of various statistical tests.
The normal distribution or Gauss distribution is defined as the distribution with the density
𝐟 𝐱 = 𝟏 𝛔 𝟐𝛑 𝐞𝐱𝐩 − 𝟏 𝟐 𝐱−𝛍 𝛔 𝟐

3. Normal Distribution The normal distribution has these features
1. μ is the mean and σ the standard deviation.
2. 𝟏/(𝛔 𝟐𝛑 ) is a constant factor that makes the area under the curve of f(x) from -∞ to ∞ equal to 1.
3. The curve of f(x) is symmetric with respect to x = μ because the exponent is quadratic. Hence for μ = 0 it is symmetric with respect to the y-axis x = 0.
4. The exponential function in the above formula goes to zero very fast — the faster the smaller the standard deviation σ is, as it should be.

4. Normal Distribution
Density of the normal distribution with μ = 0 for various values of σ

5. Normal Distribution
The normal distribution has the distribution function
𝐅 𝐱 = 𝟏 𝛔 𝟐𝛑 −∞ 𝐱 𝐞𝐱𝐩 − 𝟏 𝟐 𝐱−𝛍 𝛔 𝟐 𝐝𝐯
For the corresponding standardized normal distribution with mean 0 and standard deviation 1 we denote F(x) by Ф(z). Then we simply have
Ф 𝒙 = 𝟏 𝟐𝝅 −∞ 𝒙 𝒆 − 𝒖 𝟐 \𝟐 𝒅𝐮

6. 𝐏 𝐚<𝐗≤𝐛 =𝐅 𝐚 −𝐅 𝐛 =Ф 𝐛−𝛍 𝛔 −Ф( 𝐚−𝛍 𝛔 ).
Normal Distribution
The distribution function of the normal distribution is related to the standardized distribution function by the formula
𝐅 𝐱 =Ф( 𝐱−𝛍 𝛔 ).
The probability that a normal random variable X with mean μ and standard deviation σ assume any value in an interval is
𝐏 𝐚<𝐗≤𝐛 =𝐅 𝐚 −𝐅 𝐛 =Ф 𝐛−𝛍 𝛔 −Ф( 𝐚−𝛍 𝛔 ).

7. Normal Distribution
In practical work with the normal distribution it is good to remember that about 2/3 of all values of X to be observed will lie between μ ± σ, about 95 % between μ ± 2σ, and practically all between the three-sigma limits μ ± 3σ.
𝑷 𝝁−𝝈<𝑿≤𝝁+𝝈 ≈𝟔𝟖 %,
𝑷 𝝁−𝟐𝝈<𝑿≤𝝁+𝟐𝝈 ≈𝟗𝟓.𝟓 %,
𝑷 𝝁−𝟑𝝈<𝑿≤𝝁+𝟑𝝈 ≈𝟗𝟗.𝟕 %,
Formulas are illustrated in Fig.

8. Normal Distribution
Formulas are illustrated in Fig.

9. The Exponential Distribution
The exponential distribution describes the situation wherein the hazard rate is constant. A Poisson process generates a constant hazard rate. The pdf is
This is an important distribution in reliability work, as it has the same central limiting relationship to life statistics as the normal distribution has to non-life statistics. It describes the constant hazard rate situation.

10. The Exponential Distribution
As the hazard rate is often a function of time, we will denote the independent variable by t instead of x. The constant hazard rate is denoted by λ. The mean life, or mean time to failure (MTTF), is 1/λ. The pdf is then written as

11. The Gamma Distribution
In statistical terms the gamma distribution represents the sum of n exponentially distributed random variables. The gamma distribution is a flexible life distribution model that may offer a good fit to some sets of failure data. In reliability terms, it describes the situation when partial failures can exist, that is, when a given number of partial failure events must occur before an item fails, or the time to the a-th failure when time to failure is exponentially distributed. The pdf is

12. The Gamma Distribution
where λ is the failure rate (complete failures) and a the number of partial failures per complete failure, or
events to generate a failure. Г(a) is the gamma function
When (a – 1) is a positive integer, Г(a) = (a – 1)! This is the case in the partial failure situation. The exponential distribution is a special case of the gamma distribution, when a = 1, that is,
The gamma distribution can also be used to describe a decreasing or increasing hazard rate.

13. The Weibull Distribution
The Weibull distribution is arguably the most popular statistical distribution used by reliability engineers. It has the great advantage in reliability work that by adjusting the distribution parameters it can be made to fit many life distributions. The Weibull pdf is (in terms of time t)

14. The Weibull Distribution
The corresponding reliability function is
The hazard rate is
Mean or MTTF:
Standard deviation:
β is the shape parameter and η is the scale parameter, or characteristic life.

15. Task 1
The power transformer operates for a time T, which is a random variable and distributed according to the exponential law with the density:
𝐟 𝐭 = 𝟎 𝐚𝐭 𝐭<𝟎; 𝛌 𝐞 −𝛌 𝐚𝐭 𝐭≥𝟎,
where 𝛌=0,03 1/year.
After the time T the transformer is replaced due to increased load, damage or other causes. 15

16. Task 1 A) Find: 1) average duration of transformer’s operation;
2) probability of reliable transformer’s operation during the first 10 years;
3) the probability of transformer’s failure in the period between 10 and 20 years of operation.
B) Find the probability that in 30 years:
1) the transformer will not need to be replaced;
2) the transformer needs to be replaced twice;
3) the transformer needs to be replaced at least twice. 16

17. Task 2
The random variable U, distributed according to the normal law, is a measurement error of the voltage. An allowed systematic error in the direction of overestimating is 1.2 V. The mean square deviation of the measurement error is 0.8 V. Find the probability that the deviation of the measured value from the true value will not exceed the absolute value 1.6 V.
Find the same probability if there is no systematic measurement error. 17

18. Task 3
The random value of load current I is distributed according to the normal law with the mathematical expectation of the load m = 250 A and the mean square deviation σ = 50 A. Find the probability that the real load exceeds the value 350 A. 18

19. Task 4
The load of the workshop S is a random variable with a normal distribution. Find the mathematical expectation ms if the standard deviation σs of the load is 20 kVA and the probability that real load exceeds the value 140 kVA is 19