Variant 17
1. A wardrobe mistress has given out simultaneously tickets to three persons given their coats in a wardrobe. After that she has mixed up all coats and hung up them at random. Find the probabilities of the following events: a) the wardrobe mistress will give out to each of three persons his own coat; b) only one person will receive its coat (wardrobe mistress – гардеробщица; to mix up – перепутать; to hang up – повесить что-либо; ticket – номерок).
2. Details from two automatic devices are delivered for an assemblage. It is known that the first automatic device gives 0,5 % of spoilage, and the second – 0,6 %. Find the probability that a randomly taken detail for an assemblage will be defective if 200 details have been delivered from the first automatic device, and 300 – from the second (assemblage – сборка).
3. 3 of 100 products on the average have a defect at a given technological process. Determine the probability that among randomly chosen 20000 products: a) exactly 562 products will be defective; b) from 545 up to 605 products will be defective.
4. Two independent random variables X and Y are given by the following tables of distribution:
X |
2 |
3 |
|
Y |
1 |
2 |
3 |
P |
0,7 |
0,3 |
|
P |
0,4 |
0,3 |
0,3 |
Compose the table 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 integral function:
Find: a) the differential function f(x); b) the mathematical expectation and the dispersion of X; c) the probability of hit of the random variable X into the interval (–1/2; 1/2).
6. The dispersion of a random variable distributed under a normal law is equal to 9 m, and the mathematical expectation is equal to 25 m. Find boundaries in which one should expect a value of the random variable with probability 0,9.
7. It has been established by checking the quality of produced electric bulbs that 97% of them have no less than a guaranteed lifetime (in hours). Estimate the probability that the part of bulbs (in a batch of 1000 electric bulbs) with the lifetime less than the guaranteed lifetime will differ from the probability of such an electric bulb no more than on 0,05.
8. At inspecting 60 fish packings of a given batch the following data on weight of a separate packing (in grammes) have been obtained:
248 |
239 |
263 |
242 |
279 |
246 |
265 |
242 |
263 |
255 |
273 |
249 |
209 |
252 |
266 |
278 |
247 |
248 |
202 |
252 |
259 |
235 |
249 |
264 |
246 |
275 |
274 |
296 |
232 |
217 |
292 |
294 |
223 |
265 |
267 |
258 |
213 |
277 |
275 |
266 |
248 |
286 |
268 |
288 |
232 |
262 |
244 |
246 |
252 |
259 |
258 |
254 |
231 |
251 |
229 |
258 |
267 |
249 |
253 |
249 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 200, 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.