2022 Statistics Paper - I

Suppose X1, X2, X3 and X4 are independent where X1, X2, X3 has a Bernoulli (1/3) distribution and X4 follows B (3, 2/3). Then the distribution of X1 + X2 + X3 + 3 - X4 is :

A large number of candidates from three States A, B and C appear at a selection test. Suppose 10 candidates are selected independently out of which X, Y, Z are respectively from A, B and C. If a candidate from State A had twice as much chance as one from B or from C to be selected, then cov(Y, Z) equals :

Suppose {Xn : n ≥ 1} is a sequence of independent random variables, and for each n ≥ 1, X takes the values - 3n/2, -n, n/2 and n with probability 1/4 each. Then the sequence Yn = (1 / n^α) Σ (from i=1 to n) (xi / i) converges in probability to 0 if and only if :

A certain city receives three newspapers A, B and C. Newspaper A has 50% of the readers in that city, newspaper B has 30% of the readers and newspaper C has the remaining 20% of the readers. Assuming that no one in that city reads more than one paper, the probability that among 6 randomly chosen readers in that city, 3 will read newspaper A, 2 will read newspaper B and 1 (one) will read newspaper C, is :

If X is a Poisson random variable with P(X = 1) = P(X = 2), then what is V(X) equal to ?

If X has binomial distribution with n = 3 and p = 1/3, then what is the value of E((X - 1)³) ?