Variant 7
1. A buyer has bought a TV set and a video recorder. The probability that a TV set will sustain its warranty period is 0,9 and a video recorder – 0,85. Find the probability that: a) both devices will sustain their warranty periods; b) at least one of them will sustain the own warranty period (to sustain a warranty period – выдержать гарантийный срок).
2. Assume that 40% of all men and 30% of all women are short-sighted persons. A randomly chosen person is short-sighted. Assume that the numbers of men and women are equal. What is the probability that this person is a woman? (short-sighted – близорукий).
3. A coming up seeds of a given grade of rice is estimated with the probability equal 0,9. What is the probability that no less than five seeds will come up from the sown six ones? (to come up – всходить; seed – семя; grade – сорт; rice – рис; to sow – сеять).
4. Two independent random variables X and Y are given by the following tables of distribution:
X |
1 |
2 |
3 |
|
Y |
3 |
4 |
P |
0,3 |
0,5 |
0,2 |
|
P |
0,6 |
0,4 |
Compose the table of distribution of the random variable Z = X Y and check the property: M(X Y) = M(X) M(Y).
5. The probability density of distribution of a random variable is given by:
Find the parameter C, the mathematical expectation, the dispersion, the distribution function of the random variable X, and the probability of the event: –2 < X < 2.
6.
A normally distributed random variable X
is given by the following density of distribution:
Find the probability of hit of the random variable X into the interval (2; 4).
7. The average value of the weight of a product is 300 grammes, and the dispersion is 20. By using the Chebyshev’s inequality estimate the probability that the weight of a randomly taken product will be no less than 280 g and no more than 320 g.
8. There are the following data on the size of revenues of 100 small banks:
212 |
221 |
222 |
240 |
232 |
224 |
230 |
209 |
225 |
222 |
223 |
201 |
255 |
232 |
235 |
238 |
222 |
253 |
238 |
226 |
218 |
259 |
205 |
237 |
208 |
249 |
245 |
250 |
232 |
228 |
249 |
229 |
212 |
217 |
212 |
210 |
226 |
222 |
218 |
225 |
232 |
218 |
245 |
222 |
209 |
229 |
225 |
242 |
228 |
220 |
202 |
212 |
215 |
228 |
228 |
229 |
252 |
225 |
242 |
228 |
218 |
220 |
212 |
215 |
243 |
218 |
242 |
229 |
218 |
236 |
206 |
226 |
240 |
216 |
233 |
205 |
223 |
242 |
228 |
232 |
212 |
240 |
203 |
222 |
218 |
225 |
233 |
224 |
211 |
216 |
219 |
223 |
244 |
238 |
230 |
231 |
226 |
234 |
232 |
216 |
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.