20.7 Mathematical statistics

Problem: Consider a random sample \( X_{i} \) from some known distribution and answer the following questions: a) find the estimate of the parameter \( A \) using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-1 ; A) \); b) find the estimate of the parameter \( B \), using the method of moments, if it is known that the sample is made from the uniform distribution \( U(-B ; B) \); c) find the estimates of parameters \( c \) and \( C \) using the method of maximum likelihood estimation, if it is known that the sample is made from the uniform estimation \( U(c ; C) \); d) find (and compare) the estimates of the parameter \( L \) using the method of moments and the method of maximum likelihood estimation, if it is known that the sample is made from the exponential distribution \( E_{L} \); e) find the estimate of the parameter \( m \) using the method of moments, if it is known that the sample is made from the normal distribution \( N(m, 1) \); f) find the estimates of the parameters \( M \) and \( S \) using any known method, if it is known that the sample is made from the normal distribution \( N(M, S) \); g) plot a histogram and a polygon based on the sample, the number of intervals is 3 ; h) in each of the points (a) - (f) estimate the proximity of this theoretical distribution to the empirical one