- Consider 4 Independent Rolls Of A Fair 6 Sided Die Let X Be The Number Of 1 S And Let Y Be The Number Of 2 S Obtained 1 (145.41 KiB) Viewed 35 times
i need detail soltion how they come
Consider 4 independent rolls of a fair 6-sided die. Let X be the number of 1's and let Y be the number of 2's obtained. a) Use the definition to compute the marginal pmfs px (x) and py (x). (Hint: Which of the four important discrete random variables are X and Y? Uniform, Bernoulli, Binomial, or Poisson?) b) Compute the conditional pmf px|y. c) Compute p(x, y), the joint pmf of X and Y using your solutions from (a) and (b). Answers S Statistics and Probability expert a) The number of 1's and the number of 2's in n = 4 rolls of a 6-sided die have Binomial distribution with n = 4, p = 1/6. The PMF of X is px(x) = P(X = x) = (*) (1/6)* (5/6)¹–×‚× = 0, 1, 2,..., 4 The PMF of Y is Py(y) = P (Y = y) = (*)(1/6) (5/6)¹-v ; y = 0,1,2, ..., 4 b) The conditional PMF is 4-y Px|y= = P(X = x] Y = y) = (ª + V) (1/6) (5, 5) (5/6)4-y-x ; x = 0,1,2,...,4-y; y = 0, 1, 2, 3, 4 X c) Using the property of joint and conditional PMFs, PX|Y = PX|YPY(y) = P(X=x|Y = y) P(Y = y) 4-y-x Px|y = ~= (4xY) (1/6) * (5/6) ¹ * * (*) (1/6) (5/6)¹-v X p(x, y) = (4x) (+) (1/6)x+y (5/6)8-2y-x ; x = 0, 1, 2, ..., 4-y; y = 0,1,2,3,4