An Introduction to Probability and Statistical Inference, 2nd Edition

Section 4.5

5.2 If X₁, X₂, and X₃ are the numbers of customers buying brand A, brand B, or just browsing, then X₁, X₂, X₃ have the Multinomial distribution with parameters n = 10, p₁ = 0.25, p₂ = 0.40, and p₃ = 0.35. Therefore

(i) $P\left(X_1=2,X_2=3,X_3=5\right)=\frac{10!}{2!3!5!}{\left(0.25\right)}^2\times {\left(0.40\right)}^3\times {\left(0.35\right)}^5\simeq \mathrm{0.053.}$

(ii) $P\left(X_1=1,X_2=3|X_3=6\right)=\frac{4!}{1!3!}{\left(\frac{5}{13}\right)}^1{\left(\frac{8}{13}\right)}^3\simeq \mathrm{0.359.}$

5.4 They follow by taking the appropriate derivatives and evaluating them at 0.

5.6 The second line in (51) follows from the first line by adding and subtracting the quantity ${\rho }^2{\left(\frac{x_1-{\mu }_1}{{\sigma }_1}\right)}^2$. The expression in the ...