Question

In: Advanced Math

Probabilities Compute P(X1 + 2X2 -2X3 > 7) , where X1, X2, X3 are iid random...

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.

Expert Solution

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

Colby Messinger answered 2 years ago

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) =...

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1 then, find a two dimensional sufficient statistic for (a, b)

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if...

Let X1, X2, X3 be continuous random variables with joint pdf f(X1, X2, X3)= 2 if 1<X1<2 -1<X2<0 -X2-1<X3<0 0 otherwise Find Cov(X2, X3)

1. Compute the sample standard deviation of the following data: x1 = 12,x2 = 7,x3 =...

1. Compute the sample standard deviation of the following data: x1 = 12,x2 = 7,x3 = 6,x4 = 8,x5 = 9. A. ≈ 2.3 B. ≈ 2.6 C. ≈ 5.3 D. ≈ 6.8 E. other value 2. Suppose a simple random sample of size 16 from a normal population yields observed statistics of x¯ = 15.7 and s = 3.8. Find a 99% conﬁdence interval for the population mean µ. A. [12.81,18.59] B. [12.9,18.50] C. [12.93,18.47] D. [13.25,18.15] E. other...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having...

Consider independent random variables X1, X2, and X3 such that X1 is a random variable having mean 1 and variance 1, X2 is a random variable having mean 2 and variance 4, and X3 is a random variable having mean 3 and variance 9. (a) Give the value of the variance of X1 + (1/2)X2 + (1/3)X3 (b) Give the value of the correlation of Y = X1- X2 and Z = X2 + X3.

x1 + x2 - 2x4 = 2 x1 + x2 + 2x3 + 6x4 + x5...

x1 + x2 - 2x4 = 2 x1 + x2 + 2x3 + 6x4 + x5 = 8 −2x1 - 2x2 + x3 + 8x4 = −1 3x3 + 12x4 + 2x5 = 9 Let the linear system be given. a. Find the reduced row eelon form of the combined matrix (augmented matrix) of the system. b. Is the system consistent? If the system is consistent, find the overall solution of the system. c. Do all the solutions of the...

let X1, X2, X3 be random variables that are defined as X1 = θ + ε1...

let X1, X2, X3 be random variables that are defined as X1 = θ + ε1 X2 = 2θ + ε2 X3 = 3θ + ε3 ε1, ε2, ε3 are independent and the mean and variance are the following random variable E(ε1) = E(ε2) = E(ε3) = 0 Var(ε1) = 4 Var(ε2) = 6 Var(ε3) = 8 What is the Best Linear Unbiased Estimator(BLUE) when estimating parameter θ from the three samples X1, X2, X3

Maximize Z= 3 X1+4 X2+2.5X3 Subject to 3X1+4X2+2X3≤500 2X1+1X2+2X3≤400 1X1+3X2+3X3≤300 X1,X2,X3≥0 Change objective function coeffiecient x3...

Maximize Z= 3 X1+4 X2+2.5X3 Subject to 3X1+4X2+2X3≤500 2X1+1X2+2X3≤400 1X1+3X2+3X3≤300 X1,X2,X3≥0 Change objective function coeffiecient x3 to 6 and change coefficient of x3 to 5in constraint 1 ,to 2 in constraint 2 ,to 4 in constraint3. calculate new optimal solution using sensitivity analysis

Suppose X1, X2, . . . are a sequence of iid uniform random variables with probability...

Suppose X1, X2, . . . are a sequence of iid uniform random variables with probability density function f(x) = 1 for 0 < x < 1, or 0, otherwise. Let Y be a continuous random variable with probability density function g(y) = 3y2 for 0 < y < 1, or 0, otherwise. We further assume that Y and X1, X2, . . . are independent. Let T = min{n ≥ 1 : Xn < Y }. That is, T...

Let X1, X2, . . . be iid random variables following a uniform distribution on the...

Let X1, X2, . . . be iid random variables following a uniform distribution on the interval [0, θ]. Show that max(X1, . . . , Xn) → θ in probability as n → ∞