Question: Let X₁, X₂,…,X₁₀ be 10 independent Poisson random variables with mean 2. Let Y₁ be the number of X_i’s that are zero. Let Y₂ be the number of the X_i’s that are one. Let Y₃ be the number of the X_i’s that are more than one.

(a) Give the joint probability mass function of Y₁, Y₂, and Y₃, and determine

P(Y₁ = 1, Y₂ = 3, Y₃ = 6)

(b) Specify the marginal distribution of Y₂.

(c) Determine P(Y₁ = 2 | Y₃ = 6)

(d) Determine P(Y₂ = 3, Y₃ = 6)