Random variables [math]X[/math] and [math]Y[/math] have joint distribution

	X=0	X=1	X=2
Y=0	1/15	a	2/15
Y=1	a	b	a
Y=2	2/15	a	1/15

Let [math]a[/math] be the value that minimizes the variance of [math]X.[/math]

Calculate the variance of [math]Y[/math].

• 2/5
• 8/15
• 16/25
• 2/3
• 7/10

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

The marginal distribution of X has probability 1/5 + a at 0, 2a + b at 1, and 1/5 + b at 2. Due to symmetry, the mean is 1 and so the variance is (0 − 1) 2 (1/ 5 + a ) + (1 − 0) 2 (1/ 5 + a )= 2 / 5 + 2a which is minimized at a = 0. The marginal distribution of Y is the same as that of X and thus has the same variance, 2/5 + 0 = 2/5.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.