Variant 21
1. Participants of a toss-up pull tickets with numbers from 1 up to 110 from a box. Find the probability that the number of the first randomly taken ticket does not contain the digit 6 (toss-up – жеребьевка; ticket – жетон).
2. Products are delivered in a shop from two factories: 60% of them from the first factory and 40% – from the second. 15% of products made by the first factory are defective, and 19% of products made by the second factory are defective. Find the probability that a product bought in the shop will not be defective.
3. Let 6% of made products do not satisfy the standard in a given technological process. Determine the probability that among 120 randomly chosen products: a) exactly 113 products will satisfy the standard; b) from 98 up to 116 products will satisfy the standard.
4. Two independent random variables X and Y are given by the following tables of distribution:
X |
–1 |
3 |
|
Y |
–2 |
0 |
1 |
P |
0,6 |
0,4 |
|
P |
0,3 |
0,1 |
0,6 |
Compose the table of distribution of the random variable X – Y and check the property: D(X – Y) = D(X) + D(Y).
5. A random variable X has the following density of distribution:
Find: а) the parameter C and the distribution function F(x); b) the probability of hit of the random variable X into the interval (1,5; 4).
6. Results of measuring the distance between two cities are subordinated to a normal law with mathematical expectation 200 km and dispersion 16. Find the probability that the distance between these points is: a) no less than 190 km; b) no more than 205 km; c) from 195 up to 203 km.
7. The dispersion of each of pairwise independent random variables does not exceed 15. It is required to determine how many such random variables should take in order to assert with probability no less than 0,95 that the absolute value of the deviation of the arithmetic mean of these variables from the arithmetic mean of their mathematical expectations will not exceed 0,4.
8. There are the following data on size of annual charges of 50 workers of an enterprise on communication services:
115 |
131 |
132 |
174 |
192 |
149 |
135 |
155 |
147 |
174 |
143 |
147 |
163 |
149 |
117 |
173 |
132 |
112 |
146 |
146 |
123 |
124 |
162 |
152 |
102 |
125 |
122 |
172 |
108 |
136 |
144 |
151 |
166 |
152 |
136 |
104 |
124 |
144 |
182 |
134 |
150 |
148 |
142 |
152 |
142 |
148 |
198 |
168 |
188 |
168 |
1) Compose the interval and the discrete variation series taking the beginning of the first interval equal 100, 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.