Arsenal and Manchester United are playing a Premier League game against one anot

Question

Arsenal and Manchester United are playing a Premier League game against one another. The number of goals per game that Arsenal scores follows a Poisson distribution with mean 1.7, while the number of goals per game that Manchester United scores also follows a Poisson distribution with mean 1.3. Assume that the number of goals Arsenal scores is independent of the number of goals that Manchester United scores. What is the probability that Arsenal scores exactly 3 goals? What is the probability that Manchester City scores no more than 1 goal? What is the probability that the game ends in a 2 - 2 tie? What is the probability that the total number of goals scores in the game is 5?

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a.
Mean (X) = 1.7
P( X = 3 ) = e ^-1.7 * 1.7^3 / 3! = 0.1496

b.
Mean (Y) = 1.7
P( Y < = 1) = P(X=1) + P(X=0)
= e^-1.3 * 1.3 ^ 1 / 1! + e^-1.3 * 3 ^ 0 / 0!
= 0.6268

c.
P( X = 2 ) = e ^-1.3 * 1.3^2 / 2! = 0.2303
P( Y = 2 ) = e ^-1.7 * 1.7^2 / 2! = 0.264

Probab end in 2-2 tie = P(X=2) + P(Y = 2) = 0.2303+.264 = 0.4943

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