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Worked solution

Count products with their multiplicities before calculating moments

An unbiased regular tetrahedral die has faces 1,2,3,4. On throwing it onto a horizontal table, the face in contact with the table gives the number. Two such dice are thrown and X is the product of their numbers. Obtain the probability distribution of X, then derive

(i) P(X>8)P(X>8),

(ii) E(X)E(X),

(iii) Var⁡(X)\operatorname{Var}(X).

[1982/II/11]

CJC · Discussion · Question 3

Answer

Distribution above; (i) 1/41/4; (ii) 25/425/4; (iii) 275/16275/16.

Retain all sixteen cells of the source product table

Using independent dice, all sixteen ordered pairs are equally likely. Rows are the first die, columns the second.

First / second1234
11234
22468
336912
4481216
How did this part go?

Group equal products into the full distribution

xP(X=x)P(X=x)
11/161/16
22/162/16
32/162/16
43/163/16
62/162/16
82/162/16
91/161/16
122/162/16
161/161/16

These nine products are not equally likely;4 has three contributing cells.

How did this part go?

(i) Strictly above 8

P(X>8)=P(X=9)+P(X=12)+P(X=16)=1/16+2/16+1/16=1/4P(X>8)=P(X=9)+P(X=12)+P(X=16)=1/16+2/16+1/16=1/4.

How did this part go?

(ii) Retain all nine expectation terms

E(X)=1(1/16)+2(2/16)+3(2/16)+4(3/16)+6(2/16)+8(2/16)+9(1/16)+12(2/16)+16(1/16)=25/4E(X)=1(1/16)+2(2/16)+3(2/16)+4(3/16)+6(2/16)+8(2/16)+9(1/16)+12(2/16)+16(1/16)=25/4.

How did this part go?

(iii) Second moment and variance

E(X2)=12(1/16)+22(2/16)+32(2/16)+42(3/16)+62(2/16)+82(2/16)+92(1/16)+122(2/16)+162(1/16)=225/4E(X^2)=1^2(1/16)+2^2(2/16)+3^2(2/16)+4^2(3/16)+6^2(2/16)+8^2(2/16)+9^2(1/16)+12^2(2/16)+16^2(1/16)=225/4. Hence Var⁡(X)=E(X2)−[E(X)]2=225/4−(25/4)2=275/16\operatorname{Var}(X)=E(X^2)-[E(X)]^2=225/4-(25/4)^2=275/16.

How did this part go?

Common mistakes

Square each possible product before averaging for the second moment. Squaring the mean gives a different quantity.

Solution notes

Initial distribution ask and all three numbered asks retained. Every source product-table cell, distribution entry and moment term retained; source independent-dice model stated. No marks printed.

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