Question

poisson random variable reference table practice worksheet - statistics - ahs - 2025

1. the new york red bulls soccer team averages 1.8 goals per game. let ( x = ) the number of goals

scored in a randomly selected game. find:
a. ( p(x = 3) )
b. ( p(xleq1) )
c. ( p(xgeq3) )
d. ( p(x = 2) )
e. ( p(x < 4) )
f. ( p(x>0) )

Explanation:

Step1: Recall Poisson formula

The probability mass function of a Poisson random variable \(X\) with parameter \(\lambda\) is \(P(X = k)=\frac{e^{-\lambda}\lambda^{k}}{k!}\), and for cumulative probabilities, we sum up the relevant \(P(X = k)\) values. Here \(\lambda = 1.8\).

Step2: Calculate \(P(X = 3)\)

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Using a calculator, \(e^{-1.8}\approx0.1653\), so \(P(X = 3)=\frac{0.1653\times5.832}{6}\approx0.1607\)

Step3: Calculate \(P(X\leq1)\)

\(P(X\leq1)=P(X = 0)+P(X = 1)\)

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Step4: Calculate \(P(X>3)\)

\(P(X>3)=1 - P(X\leq3)\)

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Step5: Calculate \(P(X = 2)\)

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Step6: Calculate \(P(X<4)\)

\(P(X<4)=P(X\leq3)\approx0.8916\) (calculated in Step 4)

Step7: Calculate \(P(X>0)\)

\(P(X>0)=1 - P(X = 0)\)
Since \(P(X = 0)=e^{-1.8}\approx0.1653\), then \(P(X>0)=1 - 0.1653=0.8347\)

Answer:

A. \(P(X = 3)\approx0.1607\)
B. \(P(X\leq1)\approx0.4628\)
C. \(P(X>3)\approx0.1084\)
D. \(P(X = 2)\approx0.2681\)
E. \(P(X<4)\approx0.8916\)
F. \(P(X>0)\approx0.8347\)