Question

In: Statistics and Probability

Suppose you have decided to take up parachuting, and will land at a random point on a street (called F Avenue) between University X and University Y.

(a) You’ll be on time to class if you land closer to University X than to University Y. Find the probability that you are on time to class.

(b) What’s the probability that your distance to University X is more than three times your distance to University Y?

(c) You’ve managed to convince your two friends, who are also taking a class at 10:30, to start parachuting to school with you. If you operate independently, what’s the probability that exactly one of the three of you lands closer to University X than to University Y, and makes it to class on time?

Expert Solution

Let the distance between X and Y be . Let the distance where you lands from university X be the random variable whose PDF is uniform given by,

a) The probability that you land closer to University X than to University Y is

b) When your distance to University X is equal to three times your distance to University Y then

The required probability is

c) Use part (a). The probability that one of you lands closer to University X than to University Y is .

Now use Binomial distribution. The the number of students out of 3 who lands closer to University X than to University Y has follows Binomial distribution with
The PMF of is .

The required probability is

orchestra answered 1 year ago

(a) Suppose that the tangent line to the curve y = f (x) at the point...

(a) Suppose that the tangent line to the curve y = f (x) at the point (−9, 53) has equation y = −1 − 6x. If Newton's method is used to locate a root of the equation f (x) = 0 and the initial approximation is x1 = −9, find the second approximation x2. (b) Suppose that Newton's method is used to locate a root of the equation f (x) = 0 with initial approximation x1 = 9. If the...

Suppose that we have two random variables (Y,X) with joint probability density function f(y,x). Consider the...

Suppose that we have two random variables (Y,X) with joint probability density function f(y,x). Consider the following estimator of the intercept of the Best Linear Predictor: A = ?̅ - B • ?̅ , where ?̅ is the sample mean of y, ?̅ is the sample mean of x, and B is the sample covariance of Y and X divided by the sample variance of X. Identify the probability limit of A (if any). For each step in your derivation,...

Let the random variable X and Y have the joint pmf f(x, y) = , c...

Let the random variable X and Y have the joint pmf f(x, y) = , c xy 2 where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are {(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} . (a) Find c > 0 . (b) Find μ . X (c) Find μ . Y (d) Find σ . 2 X (e) Find σ . 2 Y (f) Find Cov (X, Y )...

Suppose X and Y are random variables with joint density f(x, y) = c(x2y + y2),...

Suppose X and Y are random variables with joint density f(x, y) = c(x2y + y2), − 1 ≤ x ≤ 1, 0 ≤ y ≤ 1 (0 else). a) Find c. b) Determine whether X and Y are independent. c) Compute P(3X + 2Y > 1 | −1/2 ≤ X ≤ 1/2).

Suppose X and Y have joint probability density function f(x,y) = 6(x-y) when 0<y<x<1 and f(x,y)...

Suppose X and Y have joint probability density function f(x,y) = 6(x-y) when 0<y<x<1 and f(x,y) = 0 otherwise. (a) Indicate with a sketch the sample space in the x-y plane (b) Find the marginal density of X, fX(x) (c) Show that fX(x) is properly normalized, i.e., that it integrates to 1 on the sample space of X (d) Find the marginal density of Y, fY(y) (e) Show that fY(y) is properly normalized, i.e., that it integrates to 1 on...

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x...

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x = 1, 2, 3 ; y = 1, 2, x+y<= 4, that is (x,y) are {(1,1), (1,2), (2,1), (2,2), 3,1)} (a) Find . c > 0 (b) Find . μX (c) Find . μY (d) Find . σ2 X (e) Find . σ2 Y (f) Find Cov . (X,Y ) (g) Find p , Corr . (x,y) (h) Are and X and Y independent

let the random variables x and y have the joint p.m.f f(x,y)= (x+y)/12 where x=1,2 and...

let the random variables x and y have the joint p.m.f f(x,y)= (x+y)/12 where x=1,2 and y=1,2 calculate the covariance of x and y calculate the correlation coefficient of x and y Thank you

Suppose that X and Y have the following joint probability density function. f (x, y) =...

Suppose that X and Y have the following joint probability density function. f (x, y) = (3/394)*y, 0 < x < 8, y > 0, x − 3 < y < x + 3 (a) Find E(XY). (b) Find the covariance between X and Y.

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9 in...

Suppose that the random variables, ξ, η have joint uniform density f(x, y) = 2/9 in the triangular region bounded by the lines x = -1 , y - -1 and y = 1 - x. a) Find the marginal densities f(x) =∫ 2/9 dy (limits, -1 to 1-x) and f(y) =∫ 2/9 dx (limits -1 to 1-y). Also show that f(x) f(y) ≠ f(x, y) so that ξ and η are not independent. b) Verify that μξ = ∫...

let the continuous random variables X and Y have the joint pdf: f(x,y)=6x , 0<x<y<1 i)...

let the continuous random variables X and Y have the joint pdf: f(x,y)=6x , 0<x<y<1 i) find the marginal pdf of X and Y respectively, ii) the conditional pdf of Y given x, that is fY|X(y|x), iii) E(Y|x) and Corr(X,Y).