2 interactive labs · AS & A Level Mathematics 9709
Poisson Events and the Interval Mean
6.1 The Poisson distribution · Scale a rate to the interval, read Poisson probabilities from the bars, add two independent streams and test whether a situation is really Poisson.
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About this lab
Scale a rate to the interval, read Poisson probabilities from the bars, add two independent streams and test whether a situation is really Poisson.
You will be able to
Calculate the mean m for an interval from a rate (3 per hour is 1.5 in 30 minutes and 4.5 in 90 minutes) and find P(X = r), P(X ≤ r), P(X < r), P(X ≥ r) and P(X > r) for X ~ Po(m) from e^(−m) m^r ÷ r!, to 3 s.f.
State that E(X) = Var(X) = m for a Poisson variable, and that independent Poisson variables add: X + Y ~ Po(m₁ + m₂), so the standard deviation is √(m₁ + m₂), not √m₁ + √m₂.
Decide whether a Poisson model suits a situation by testing the four conditions: events occur randomly, independently, singly and at a constant average rate.
Approximating Binomial and Poisson Distributions
6.1 The Poisson distribution · Decide whether Po(np) or N(m, m) is appropriate, then compare exact and approximate probabilities and see the error, with a continuity correction.
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About this lab
Decide whether Po(np) or N(m, m) is appropriate, then compare exact and approximate probabilities and see the error, with a continuity correction.
You will be able to
State and apply the conditions for approximating B(n, p) by Po(np), n > 50 and np < 5 (approximately), and compare the approximate probability with the exact binomial value.
Approximate X ~ Po(m) by N(m, m) when m > 15 (approximately), apply the continuity correction to P(X = r), P(X ≤ r), P(X < r), P(X ≥ r) and P(X > r), and read Φ(z) from the table.
Explain why an approximation fails when its conditions are not met, using the size of the error, and why leaving out the continuity correction gives P(X = r) = 0.
