Variant 19
1. Two dice are tossed. Find the probabilities of the following events:
a) the sum of aces is equal to 7;
b) the product of aces is equal to 10;
c) the sum of aces is less than their product.
2. The first automatic device makes 30 %, the second – 25 %, the third – 45 % of all nails. The spoilage makes 6, 5 and 3% respectively in their production. A randomly chosen nail has been defective. Which of automatic devices (1-st, 2-nd or 3-rd) has the nail been most likely made (nail – гвоздь)?
3. The probability that a product of a given batch does not satisfy the standard is equal to 0,3. Determine the probability that among randomly selected 500 products:
a) exactly 140 products do not satisfy the standard;
b) from 145 up to 160 products are non-standard.
4. The table of distribution of a random variable X has the following form:
X |
–1 |
0 |
2 |
4 |
5 |
Р |
0,35 |
0,15 |
0,2 |
0,12 |
0,18 |
Find the mathematical expectation and the dispersion of the random variable X.
5. The density of distribution of a random variable X is given by:
Find: a) the parameter C; b) the mathematical expectation and the dispersion of the random variable.
6. A random variable X is distributed under a normal law. The mathematical expectation and the mean square deviation of the random variable are 40 and 12 respectively. Find: a) the probability that the random variable X takes on a value belonging to the interval (20; 55); b) the probability that the absolute value of the deviation X – M(X) is less than 7.
7. By using the Chebyshev’s inequality estimate the probability that a random variable X will differ from its mathematical expectation M(X) no more than on 6 if the mean square deviation of the random variable X is equal to 3.
8. There are the following data on annual volume of confectionery products (in thousands of tons) of 60 commercial shops:
432 |
487 |
752 |
361 |
552 |
364 |
779 |
667 |
358 |
546 |
558 |
653 |
383 |
431 |
756 |
450 |
565 |
647 |
458 |
577 |
453 |
447 |
551 |
670 |
340 |
554 |
653 |
326 |
713 |
455 |
606 |
586 |
473 |
756 |
401 |
669 |
562 |
739 |
649 |
777 |
448 |
466 |
653 |
551 |
678 |
766 |
352 |
463 |
553 |
788 |
768 |
316 |
431 |
574 |
345 |
749 |
466 |
580 |
395 |
493 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 300, and the width of each interval equal 50.
2) Construct the histogram and the polygon of relative frequencies of distribution.
3) Find the mode and the median (using the discrete series).
4) Find empirical functions of distribution of continuous and discrete variation series; and construct their graphs.