Question

Let \( 0, 1, 0, 0, 1 \) be the observed values of a random sample of size five from a discrete distribution with the probability mass function \( P(X = 1) = 1 - P(X = 0) = 1 - e^{-\lambda} \), where \( \lambda>0 \). The method of moments estimate (round off to 2 decimal places) of \( \lambda \) equals .............

Hint:

The method of moments involves equating sample moments to population moments to estimate parameters. Here, the first moment (mean) was used to estimate \( \lambda \).

Answer 1

Correct Answer: 0.45

Step 1: Use the method of moments.
The method of moments estimates the parameter \( \lambda \) by equating the sample mean with the population mean. First, calculate the sample mean: \[ \text{Sample mean} = \frac{0 + 1 + 0 + 0 + 1}{5} = \frac{2}{5} = 0.4. \] Step 2: Find the population mean.
The expected value of \( X \) for a Bernoulli distribution with parameter \( P(X = 1) = 1 - e^{-\lambda} \) is: \[ E(X) = 1 - e^{-\lambda}. \] Equating the sample mean with the population mean gives: \[ 0.4 = 1 - e^{-\lambda}. \] Solving for \( \lambda \): \[ e^{-\lambda} = 0.6 \quad \Rightarrow \quad \lambda = -\ln(0.6) \approx 0.5108. \] Final Answer: \[ \boxed{0.51}. \]