use poisson distribution to show that
$e(x) = 1/\lambda$
$var(x) = 1/\lambda$

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Poisson Distribution", "Expected Value", "Variance Derivation" ], "new_concepts": [], "current_concepts": [ "Poisson Distribution", "Expected Value", "Variance Derivation" ] } ## Define the Poisson probability mass function Using the *Poisson Distribution* knowledge point $$ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}, \quad k = 0, 1, 2, \dots $$ ## Derive the expected value Using the *Expected Value* knowledge point $$ \begin{aligned} E(X) &= \sum_{k=0}^{\infty} k \frac{e^{-\lambda} \lambda^k}{k!} \ &= \sum_{k=1}^{\infty} \frac{e^{-\lambda} \lambda^k}{(k-1)!} \ &= \lambda e^{-\lambda} \sum_{m=0}^{\infty} \frac{\lambda^m}{m!} \ &= \lambda e^{-\lambda} e^{\lambda} = \lambda \end{aligned} $$ ## Derive the variance Using the *Variance Derivation* knowledge point $$ \begin{aligned} E(X(X-1)) &= \sum_{k=0}^{\infty} k(k-1) \frac{e^{-\lambda} \lambda^k}{k!} \ &= \lambda^2 e^{-\lambda} \sum_{k=2}^{\infty} \frac{\lambda^{k-2}}{(k-2)!} \ &= \lambda^2 e^{-\lambda} e^{\lambda} = \lambda^2 \ E(X^2) &= E(X(X-1)) + E(X) = \lambda^2 + \lambda \ \text{Var}(X) &= E(X^2) - [E(X)]^2 = \lambda^2 + \lambda - \lambda^2 = \lambda \end{aligned} $$ ## Address the handwritten formula discrepancy The handwritten prompt asks to show $E(X) = \lambda$ and $\text{Var}(X) = \lambda$, where the symbol written as $1/\lambda$ or $1/x$ is a common handwriting style for the Greek letter lambda ($\lambda$). $$ \begin{aligned} E(X) &= \lambda \ \text{Var}(X) &= \lambda \end{aligned} $$ For a random variable $X$ following a Poisson distribution with parameter $\lambda$: $$ E(X) = \lambda $$ $$ \text{Var}(X) = \lambda $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Statistics", "Poisson Distribution" ] }

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Question

Explanation:

Define the Poisson probability mass function

Derive the expected value

Derive the variance

Address the handwritten formula discrepancy

Answer: