V – variant number in the group list.
Individual task ¹ 2 on probability theory.
Theme:Laws of distribution and numerical characteristics of random variables.
V – variant number in the group list.
Problem 1. In eachof independent trials event occurs with constant probability . Calculate all probabilities , where – is frequency of event . Construct the graph of probabilities . Find the most probable frequency.
Values of parametres calculate with following formulas:
.
Problem 2. In eachof independent trials event occurs with constant probability . Find probability that event occurs:
1) precisely times;
2) less than and more than times;
3) more than times.
; ; ; .
Problem 3. In each of independent trials event occurs with constant probability . Find probability that event occurs:
1) Precisely times;
2) Precisely times;
3) less than and more than times;
4) less than times.
; ; ; ;
; ; .
Problem 4.At telephone station wrong connection occurs with probability . Find probability that among connections takes place:
1) Precisely wrong connections;
2) less than wrong connections;
3) more than wrong connections.
; ; ;* ;
Note, that remainder is the value, left after division (23/5=4*5+3; so, 3 is remainder; 35/5=7*5+0; 0 is remainder; 3/5=0*5+3; 3 is remainder)
; ; .
Problem 5. In eachof independent trials event occurs with constant probability . Find the probability that relative frequency of this event differs from probability no more than on
; ; ; .
Problem 6.The Random variable is set by the distribution law
Find the function of distribution of a random variable ; construct its graph. Calculate , , .
, , ,
, , ,
Problem 7.TheRandom variable is set by a function of probability density
Find function of distribution . Build graphs of ; . Calculate , , , .
,
Problem 8.TheRandom variable is set by distribution function
Find the function of probability density .Build graphs of . Calculate , , , .
Problem 9.random variable is given. Find probability that this random variable accepts value:
1) In an interval ;
2) less than ;
3) more for ;
4) Differs from the average value no more than on .
, ,
, , , ,
Problem 10.It is given:therandom variable and points , on the numerical axis which divide it into six intervals. Find the probability that this random variable accepts value in these intervals.
, , ,
, , , , ,
Date: 20151211; view: 352
