Question

3. A web-based store is stuck with a very large set of three kinds of unsold Christmas snow globes: Globes with a Nutcracker figurine, globes with a Santa figurine, and globes with a Snowman (see pictures below from left to right).

3. A web-based store is stuck with a very large set of three kinds of unsold Christmas snow globes: Globes with a Nutcracker figurine, globes with a Santa figurine, and globes with a Snowman (se pictures below from left to right). Fifty percent of the snow globes have the Nutcracker, 20% have Santa, and 30% have the Snowman. The store runs a special where they offer 10 randomly chosen snow globes for $100 a roughly "i0 for the price of 4" special (3 pts) Define random variables for each of the types of snow globes and give the name and parameters of the joint distribution of these random variables a) b) (3 pts) Find the probability of getting 3 Nutcracker snow globes and 7 Snowman snow globes in a set of 10 snow globes. Show all work. c) (6 pts) Let Xthree times the number of nutcrackers plus two times the number of snowmen. Find the mean and variance of X

Answer 1

$(a)~Let~X_1=random~variable~denoting~number~of~getting~nutcrackers\\ X_2=random~variable~denoting~number~of~getting~Santa\\ X_3=random~variable~denoting~number~of~getting~Snowman\\ Then\\ ~(X_1,X_2,X_3)\sim Multinomial~distribution(10,~p_1=0.5,~p_2=0.2,~p_3=0.3)\\ The~joint~pmf~of~X_1,X_2,X_3~is\\ f(x_1,x_2,x_3)=\frac{10!}{x_1!x_2!x_3!}(0.5)^{x_1}(0.2)^{x_2}(0.3)^{x_3};~x_1,x_2,x_3\geq 0\\ ~and~x_1+x_2+x_3=10\\ =0~otherwise\\ (b)~P(X_1=3,X_2=0,X_3=7)=\frac{10!}{3!0!7!}(0.5)^{3}(0.2)^{0}(0.3)^7=0.0033\\ c)~E(X)=E(3X_1+2X_3)=3E(X_1)+2E(X_3)=3*(10*0.5)+2*(10*0.3)\\ =21\\ Var(X)=Var(3X_1+2X_3)=9Var(X_1)+4Var(X_3)+12Cov(X_1,X_3)\\ =9*(10*0.5*(1-0.5))+4*(10*0.3*(1-0.3))+12*(-10*0.5*0.3)\\ =12.9$