15.5.2 Two-dimensional random variables and their characteristics

Problem: A «wrong» coin (the probability of getting "heads" is \( A \) ) is tossed for \( N \) times. The following variables are considered: \( x \) is the number of "heads", \( y \) is the number of "tails", \[ z_{1}=\frac{x}{y}, z_{2}=x+y, z_{3}=\frac{x}{z_{2}} \text {. } \] Answer the following questions about these random variables: a) describe the distributions of the random variables \( x, y, z_{1}, z_{2}, z_{3} \); find the expected values, the second moments, dispersions; b) describe the conditional distribution of the random variables \( x \mid y \); c) as a result of the \( M \)-th toss, it turned out that there are exactly \( L \) «heads», what is the probability that there will be no more than \( K \) "tails"? d) Let's find the covariance and the correlation coefficient of the variables \( x \) and \( y \); e) Find the covariance and the correlation coefficient of the variables \( x^{2} \) and \( y \); \[ A=0,69 ; N=252 ; M=142 ; L=80 ; K=55 \text {. } \]