For a Poisson random variable X with parameter λ greater than zero, the moment-generating function is M(t) = exp[λ(eᵗ − 1)]. Derive it from the expectation of eᵗˣ so that you can explain the result and use it to recover moments.
Start with the probability mass function
For non-negative integer k, the Poisson probability is P(X = k) = e⁻λ λᵏ/k!. The parameter describes the distribution; it is not the observed value k. State the support and keep the random variable, its possible values and the parameter distinct before manipulating the series.
Substitute into the expectation
Write M(t) = Σ eᵗᵏ e⁻λ λᵏ/k!, summing from k = 0 to infinity. Factor out e⁻λ and combine the remaining terms as (λeᵗ)ᵏ/k!. This step is a direct substitution into the MGF definition, not a separate assumption about the data.
Recognise the exponential series
The series Σ zᵏ/k! equals eᶻ. With z = λeᵗ, the expression becomes e⁻λe^(λeᵗ), or exp[λ(eᵗ − 1)]. Show this connection explicitly in an assignment. Skipping from the initial summation to the final answer can conceal whether you recognised the series or simply recalled the formula.
Recover and distinguish moments
At t = 0, M(0) = 1, as expected. The first derivative at zero gives E[X] = λ. The second derivative at zero gives E[X²] = λ² + λ, so Var(X) = λ. Do not confuse the second raw moment with the variance; subtract the square of the mean where the variance calculation requires it.
Keep the distribution assumption separate
Deriving the MGF does not prove that a dataset follows a Poisson distribution. A modelling question needs its own justification and checks. State the assumptions prescribed by the course and explain the interpretation of λ in the case. Use consistent notation and distinguish a symbolic identity from an estimated parameter.
Sources and further reading
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