Elements of math statistics  Examples
Examples [2]
E x a m p l e 1. Trains of underground go on a regular basis with an interval 2 minutes. Passenger leaves on platform during the casual moment of time which in any way has been not connected with the schedule. A random variable – time Т during which it should wait for a train. Find density function, mathematical expectation, dispersion, average squarelaw deviation and probability of what to wait to the passenger it is necessary no more halfminute.
S o l u t i o n . Density function f (x) =1/2 (0 < x < 2) is shown on Fig. 9.
Fig. 9. Density function
E x a m p l e 2. Random variable X has exponent distribution with parameter. Find probability of event {1 < X < 2}.
S o l u t i o n . We have The probability of hit in an interval (1, 2) is equal to an increment of function of distribution on this interval:
E x a m p l e 3. The matrix of distribution of system of two random variables X and Y is given by the table:
y_{j}
x_{i} 
0 
2

5 
1 
0.1 
0 
0.2 
2 
0 
0.3 
0 
4 
0.1 
0.3 
0 
Find numerical characteristics of system of random variables (X, Y): mathematical expectations, dispersions, average squarelaw deviations, covariance and coefficient of correlation.
S o l u t i o n . First we’ll find series of distribution of separate values entering into the system. Summarizing probabilities p_{ij}, being in the 1st, the 2nd and the 3rd lines, we’ll receive:
Series of distribution of random variable X looks like:
Mathematical expectation
From formula (8) dispersion
Average squarelaw deviation
Similarly summarizing probabilities p_{ij}, being in the 1st, the 2nd and in the 3rd columns, we’ll receive:
Series of distribution of random variable Y looks like:
Mathematical expectation
From formula (8) dispersion
Average squarelaw deviation
Covariance from formula (52) is equal to
Coefficient of correlation: thus, between random variables X and Y there is a negative linear correlation, i.e. at increase in one of them another has decreases.
