Question

Probabilities

Compute P(X₁ + 2X₂ -2X₃ > 7) , where X₁, X₂, X₃ are iid random variables with common distribution N(1,4). You must give the numerical answer.

Specify and show any formulas used for E, P, and Var

for example Var( X₁ + 2X₂ -2X₃) = 4 + 2²(4) + (-2)²(4) = 35. Please specify how all the individual numbers where obtained and the formula used.

Answer 1

Consider the following formula:

(i) For Independent and identically distributed (i.i.d.) random variables x₁, x₂ and constant a & b,

                                      Var(aX₁+bX₂) = a²Var(X₁) + b²Var(X₂)

(ii) For random variables x₁, x₂ and constant a & b,

                                       E(aX₁+bX₂) = aE(X₁) + bE(X₂)

(iii) For any real value a and continuous random variable X,

                                     P(X>a) = 1-P(x<=a)

Solution:

X₁, X₂, X₃ are i.i.d. random variables with N(1,4). So,

for each i = 1,2,3; E(x_i) = 1 and Var(X_i) = 4.

Let Y = X₁+2X₂-2X₃

Mean of Y is

E(Y) = E(X₁+2X₂-2X₃) = E(X₁) + 2E(X₂) -2E(X₃) (From (ii))

                                     = 1+2(1)-2(1) = 1

Var(Y) = Var(X₁+2X₂-2X₃) = Var(X₁) + 2²Var(X₂) + (-2)²Var(X₃)    (from (i))

                                            = 4 + 2²(4) + (-2)²4 = 36

So deviation of Y = (Var(Y))^1/2 = (36)^1/2 = 6

So, P(X₁+2X₂-2X₃>7) = P(Y>7) = 1 - P(Y<=7) = 1 - P((Y - E(Y))/(deviation of Y) <= (7 - E(Y))/(deviation of Y) )

                                                                            = 1 - P((Y - 1)/6 <= (7 - 1)/6 )

                                                                            = 1 - P(Z <= 1)    (Z = (Y - 1)/6 is a standard normal variable)

                                                                           = 1 - Phi(1)     

                                                                         = 1 - 0.8413     (From standard normal table )

                                                                           = 0.1587