Question

12. Suppose , 8. denote the proportions of Democrats (D), Republicans (R) and Others (O) in a large population of voters. (S00S0.8.25 1 anda+8+= 1.) An individual is selected at random from the population. Write X = 1, Y =0,2 = 0 if that individual is Dwrite X = 0, y = 1,2 = 0 if the individual is R and write X = 0, y = 0,2 = 1 if the individual is O. Find: a) E(X), E(Y); b) Var(X), Var(Y); c) Cov(X,Y). Suppose next that n individuals are selected independently and randomly with replace- ment from the population. The total number of D's may be written, D. = Xi+...+X.. Similarly let R = Y; + ... + Y and let 0. = 2 + ... + Z. Let D. - R. denote the excess of D's over R's selected. Find d) ED. - R.): e Var(D.-R.).

Answer 1

Solution

Back-up Theory

If a discrete random variable, X, has probability function, p(x), x = x1, x2, …., xn, then

E(X) = Σ{x.p(x)} summed over all possible values of x…..…............................................. (1)

E(X²) = Σ{x².p(x)} summed over all possible values of x…….................…………………..(2)

Variance of X = Var(X) = σ2 = E[{X – E(X)}²] = E(X²) – {E(X)}²……...........………………..(3)

Let X and Y be two random variables with joint probability p_XY(x,y). Then,

E(XY) = ∑{(x.y). p_XY(x,y)}, sum over all possible values of X and Y…............…….………(4)

Cov(X, Y) = covariance of X and Y = E(XY) – {E(X). E(Y)} ……………….....................…..(5)

E(X - Y) = E(X) - E(Y) ………………………..……………...................................……………(6)                      

Var(X - Y) = Var(X) + Var(Y) - 2Cov(X,Y)…..…………................................…..………….…(7)

If X₁, X₂, X₃, ……, X_n are iid (i.e., independently and identically distributed) with

mean µ and variance σ², and Y = (X₁ + X₂ + X₃+ …… + X_n), then

E(Y) = E{∑[1,n](X_i)} = ∑[1,n](µ) = nµ   ................................................................................(8)

Var(Y) = Var{∑[1,n](X_i)} =Var{∑[1,n](X_i)} = ∑[1,n]V(X_i)} = ∑[1,n]σ² = nσ² .......................... (9)

Now to work out the solution,

Part (a)

X = 1 if the individual is a D with probability α

    = 0 otherwise.

So, vide (1), E(X) = α Answer 1

Similarly, E(Y) = β Answer 2

Part (b)

Vide (2), E(X²) = 1² x α = α.

So, vide (3), Var(X) = α – α² = α(1 – α) Answer 3

Similarly, Var(Y) = β(1 – β)   Answer 4

Part (c)

The product xy = 1 when both x and y are 1, otherwise it is 0.

So, vide (4), E(XY) = 1 x αβ = αβ ..................................................................(10)

Vide (5), Answers 1 and 2, (10),

Cov(X,Y) = αβ – αβ = 0 Answer 5

Part (e)

Vide (8) and Answer 1,

E(Dn) = nα

Vide (8) and Answer 2,

E(Rn) = nβ

So, vide (6),

E(Dn - Rn) = n(α – β) Answer 6

Part (f)

Vide (9) and Answer 3,

Var(Dn) = nα(1 – α)

Vide (9) and Answer 4,

Var(Rn) = nβ(1 – β)

So, vide (7) and Answer 5,

Var(Dn - Rn) = n{α(1 – α) + β(1 – β)} Answer 7

DONE