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}.
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?
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