Question

In: Statistics and Probability

Let X and Y be independent Poisson random variables with parameters 1 and 2, respectively, compute...

  1. Let X and Y be independent Poisson random variables with parameters 1 and 2, respectively, compute
  1. P(X=1 and Y=2)
  2. P(X+Y>=2)
  1. Find Poisson approximations to the probabilities of the following events in 500 independent trails with probabilities 0.02 of success on each trial.
  1. 1 success
  2. 2 or fewer success.

Solutions

Expert Solution


Related Solutions

Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the...
Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the expected value E(XY ). b) What is the probability density function fZ(z) of Z = XY ? Hint: First compute the cumulative distribution function FZ(z) = P(Z ≤ z) using a double integral, and then differentiate in z. c) Use your answer to b) to compute E(Z). Compare it with your answer to a).
Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2,...
Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2, 4]). a) FindP(X<Y). b) FindP(X<Y|Y>3) c) FindP(√Y<X<Y).
Let X and Y be independent random variables with mean 𝜇X and 𝜇𝑌, and variances 𝜎𝑋 2 and 𝜎𝑌 2 respectively
  Let X and Y be independent random variables with mean 𝜇X and 𝜇𝑌, and variances 𝜎𝑋 2 and 𝜎𝑌 2 respectively. Show that 𝑉𝑎𝑟[𝑋 ∙ 𝑌] = 𝜎𝑋 2 ∙ 𝜎𝑌 2 + 𝜇𝑌 2 ∙ 𝜎𝑋 2 + 𝜇𝑋 2 ∙ 𝜎𝑌 2
Let X and Y be two independent random variables such that X + Y has the...
Let X and Y be two independent random variables such that X + Y has the same density as X. What is Y?
Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) =...
Let X and Y be jointly normal random variables with parameters E(X) = E(Y ) = 0, Var(X) = Var(Y ) = 1, and Cor(X, Y ) = ρ. For some nonnegative constants a ≥ 0 and b ≥ 0 with a2 + b2 > 0 define W1 = aX + bY and W2 = bX + aY . (a)Show that Cor(W1, W2) ≥ ρ (b)Assume that ρ = 0.1. Are W1 and W2 independent or not? Why? (c)Assume now...
Let X and Y be two independent random variables. X is a binomial (25,0.4) and Y...
Let X and Y be two independent random variables. X is a binomial (25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y. a) Find the expected value of X, the expected value of Y, the variance of X and the variance of Y. b) Find the expected value of W. c) Find the variance of W. d) Find the covariance of Z and W. d) Find the covariance of Z and W.
Let X ∼Exp(1), Y ∼Exp(2) be independent random variables. (a) What is the range of Z...
Let X ∼Exp(1), Y ∼Exp(2) be independent random variables. (a) What is the range of Z := X + Y ? (b) Find the pdf of Z. (c) Find MZ(t). (d) Let U = e Y . What is the range of U? (e) Find the pdf of U|X.
1. Let X and Y be independent random variables with μX= 5, σX= 4, μY= 2,...
1. Let X and Y be independent random variables with μX= 5, σX= 4, μY= 2, and σY= 3. Find the mean and variance of X + Y. Find the mean and variance of X – Y. 2. Porcelain figurines are sold for $10 if flawless, and for $3 if there are minor cosmetic flaws. Of the figurines made by a certain company, 75% are flawless and 25% have minor cosmetic flaws. In a sample of 100 figurines that are...
Let X and Y be independent discrete random variables with the following PDFs: x 0 1...
Let X and Y be independent discrete random variables with the following PDFs: x 0 1 2 f(x)=P[X=x] 0.5 0.3 0.2 y 0 1 2 g(y)= P[Y=y] 0.65 0.25 0.1 (a) Show work to find the PDF h(w) = P[W=w] = (f*g)(w) (the convolution) of W = X + Y (b) Show work to find E[X], E[Y] and E[W] (note that E[W] = E[X]+E[Y])
Let X ∈{1,2} and Y ∈{3,4} be independent random variables with PMF-s: fX(1) = 1 2...
Let X ∈{1,2} and Y ∈{3,4} be independent random variables with PMF-s: fX(1) = 1 2 fX(2) = 1 2 fY (3) = 1 3 fY (4) = 2 3 Answer the following questions (a) Write down the joint PMF (b) Calculate P(X + Y 6 5) and P(Y −X > 2) (c) Calculate E(XY ), E(X2Y ), E(︁X2+1 Y−2)︁ (d) Calculate the Cov(X,Y ), Cov(1−X,3Y + 2) and Var(2X −Y ) ( e*) Calculate Cov(XY,X), Cov(XY,X + Y )...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT