Answer

1. \(P(X=k) = \binom{10}{k} 0.2^{k}0.8^{10-k}\)

2. \(P(\text{no hits}) = 0.8^{10}\)

3. \(P(X>=6) \ \sum_{k=6}^{10} \binom{10}{k} 0.2^{k}0.8^{10-k}\)

4. \(E[X] = np = 2, Var(X) = np(1-p) = 1.6\) for Bernoulli distribution

5. \(Y = 2X - 3, E[Y] = 2E[X] - 3 = 1, Var(Y) = 4Var(X) = 6.4\)

6. \(Z = X^{2}, E[Z] = E[X^{2}] = Var(X) + E[X]^{2} = 5.6\)