Question

Can someone please answer these three questions ASAP?

1) A biased coin with probability of heads p, is tossed n times. Let X and Y be the total number of heads and tails, respectively. What is the correlation ρ(X, Y )?

2) Choose a point at random from the unit square [0, 1] × [0, 1]. We also choose the second random point, independent of the first, uniformly on the line segment between (0, 0) and (1, 0). The random variable A is the area of a triangle with its corners at (0, 0) and the two selected points. Find the probability density function (pdf) of A.

3) n people put their car keys in the center of a room where the keys are mixed together. Each person randomly selects one. Let Y be the number of people who can select their own key. Find the mean and variance of Y . (Hint: Use Xi = 1 if ith person has a match, and Xi = 0 otherwise.)

Answer 1

1)

cor(X,Y) = Cov(X,Y)/sqrt(Var(X) * Var(Y))

X follow binomial distribution with parameter n and p

q= 1-p

X+Y =n

E(X) = np

E(Y) = n*(1-p)

Var(X) = npq

Var(Y) = npq

Cov(X,Y) = E(XY) - E(X)E(Y)

E(XY) = E(X * (n- X)) = E(nX- X^2) = nE(X) - E(X^2)

= n * n*p - (Var(X) + E(X)^2)

= n^2*p - (npq + n^2p^2)

= n^2 * p - npq - n^2 p^2

= n^2 * pq - npq

= npq*(n-1)

hence

Cov(X,Y) = npq (n-1) - (np * nq)

= npq((n-1) - n)

= -npq

hence

correlation = -npq/(Sqrt(npq * npq)) = -1

they are perfectly negatively correlated

Please give me a thumbs-up if this helps you out. Thank you! :)

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