Let 𝑋 be a discrete Poisson random variable with parameter 𝜆. Given 𝑋=𝑥, let 𝑌 be the binomial random variable 𝖡𝗂𝗇𝗈𝗆(𝑥,𝑝), where 𝑝 is the binomial parameter.
Given 𝑋=𝑥, what are the values that 𝑌 can take?
Given that 𝑋=𝑥, the possible values that 𝑌 can take on will depend on the value of 𝑥.
For 𝑌 to be a binomial random variable with 𝑥 trials, 𝑌 can take on any value from 0 to 𝑥, inclusively.
Therefore, the values that 𝑌 can take on are {0, 1, 2, ..., 𝑥}.
Given 𝑋=𝑥, 𝑌 can take values from 0 to 𝑥, inclusive.
This is because 𝑌 is a binomial random variable with 𝑥 trials, and each trial can have a success or failure. The number of successes in 𝑛 trials can range from 0 to 𝑛, inclusive. In this case, 𝑥 represents the number of trials for 𝑌, so 𝑌 can take values from 0 to 𝑥.