Variant 25
1. There are 6 green and 9 yellow balls in the first urn, and 9 green and 6 yellow balls in the second urn. One ball is randomly taken out from each urn. Find the probability that at least one of them is green.
2. A warehouse has received two equal batches of footwear in identical packing boxes. It is known that 50 % of footwear in the first batch and 80 % of footwear in the second batch have a black color. What is the probability that a randomly taken pair of footwear in the warehouse will have a black color?
3. An automatic device is making plastic hangers. The probability that no non-standard hanger will be made for one hour is equal to 0,8. Find the probability that all hangers made for four hours will be standard (hanger – вешалка).
4. Two independent random variables X and Y are given by the following tables of distribution:
X |
2 |
3 |
4 |
|
Y |
– 3 |
– 2 |
– 1 |
P |
0,7 |
0,2 |
0,1 |
|
P |
0,3 |
0,5 |
0,2 |
Compose the law of distribution of the random variable Z = X + Y and check the property: M(X + Y) = M(X) + M(Y).
5.
A random variable
X
is given by the probability density of distribution
in the interval
,
and f
(x) =
0 outside of the interval. Find the dispersion of the random variable
X.
6. Let the lifetime of a radio tube of a given batch be a normally distributed random variable X with mathematical expectation a = 800 hours and mean square deviation = 30 hours. Determine the probability that the lifetime X of a randomly chosen tube from the batch: a) is in limits from = 830 hours up to = 950 hours; b) deviates from the mathematical expectation no more than on 20 hours by absolute value.
7. The discrete random variable X is given by the law of distribution:
-
X
2
3
4
P
0,3
0,4
0,3
By using the Chebyshev’s inequality estimate the probability that the random variable will deviate from the mathematical expectation no more than on 2,5 by absolute value.
8. The following data on strength of a steel wire of a given batch have been received after testing 100 samples:
453 |
407 |
438 |
435 |
426 |
438 |
402 |
434 |
474 |
485 |
468 |
487 |
457 |
469 |
428 |
455 |
436 |
449 |
425 |
436 |
468 |
459 |
436 |
482 |
416 |
466 |
441 |
452 |
454 |
453 |
474 |
477 |
413 |
459 |
453 |
443 |
446 |
461 |
465 |
455 |
471 |
432 |
448 |
466 |
459 |
439 |
449 |
451 |
431 |
468 |
413 |
417 |
452 |
495 |
492 |
451 |
465 |
446 |
475 |
473 |
442 |
452 |
449 |
479 |
448 |
498 |
449 |
432 |
463 |
455 |
462 |
455 |
462 |
443 |
465 |
445 |
478 |
442 |
422 |
443 |
469 |
445 |
446 |
455 |
445 |
463 |
451 |
468 |
479 |
463 |
452 |
469 |
457 |
454 |
468 |
452 |
477 |
469 |
453 |
457 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 400, and the width of each interval equal 10.
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.