Hello, I would appreciate some help in the following:
A fair die is cast at random three independent times. Let the Random Variable Xi be equal to the number of spots on ith trial, i=1,2,3. Let the RV Y be equal to max(Xi). Find the cdf of Y and show that the pmf is equal to \(\displaystyle \displaystyle\frac{y^3-(y-1)^3}{6^3}]\). Hint: Consider the cdf of Y, P(Xi<Y, i=1,2,3).
I think is reasonable to assume that the probability mass function of Xi is uniform and is given by 1/6. But then I get stuck in the derivation of the CDF of Y as P(Xi<Y)= \(\displaystyle \displaystyle \sum_{i=1}^y\frac{1}{6}=y/6.\). And then we can raise the fraction to the power of 3 because each trial has the same probability but that does not seem close to the result I need to show.
Do you think I am missing something here? Thanks in advance.
A fair die is cast at random three independent times. Let the Random Variable Xi be equal to the number of spots on ith trial, i=1,2,3. Let the RV Y be equal to max(Xi). Find the cdf of Y and show that the pmf is equal to \(\displaystyle \displaystyle\frac{y^3-(y-1)^3}{6^3}]\). Hint: Consider the cdf of Y, P(Xi<Y, i=1,2,3).
I think is reasonable to assume that the probability mass function of Xi is uniform and is given by 1/6. But then I get stuck in the derivation of the CDF of Y as P(Xi<Y)= \(\displaystyle \displaystyle \sum_{i=1}^y\frac{1}{6}=y/6.\). And then we can raise the fraction to the power of 3 because each trial has the same probability but that does not seem close to the result I need to show.
Do you think I am missing something here? Thanks in advance.