Step-by-step explanation:
To estimate the value of parameter b using the method of maximum likelihood, we need to maximize the likelihood function. The likelihood function is defined as the probability density function (PDF) evaluated at the observed data. In this case, the observed data is the average wave height of 3.61 m.
Given the continuous PDF: f(x) = 0.5b³x² exp(-bx)
The likelihood function, L(b), can be written as:
L(b) = f(x₁) * f(x₂) * ... * f(xₙ)
Since the wave heights are independent and identically distributed, we can rewrite the likelihood function as:
L(b) = ∏[f(xᵢ)]
Taking the natural logarithm (ln) of the likelihood function, we have:
ln(L(b)) = ∑[ln(f(xᵢ))]
To find the maximum likelihood estimate (MLE) of b, we differentiate the logarithm of the likelihood function with respect to b, set it equal to zero, and solve for b.
∂[ln(L(b))]/∂b = ∂[∑(ln(f(xᵢ)))]/∂b = 0
To simplify the calculation, we can work with the natural logarithm of the PDF:
ln(f(x)) = ln(0.5b³x² exp(-bx)) = ln(0.5) + 3ln(b) + 2ln(x) - bx
Now, we can differentiate ln(f(x)) with respect to b:
∂[ln(f(x))]/∂b = 3/b - x
Setting this derivative equal to zero, we have:
3/b - x = 0
Solving for b, we find:
b = 3/x
To estimate the value of b, we need to substitute the observed average wave height, x = 3.61 m, into the equation:
b = 3/3.61 ≈ 0.830
Therefore, the estimated value of the unknown parameter b using the method of maximum likelihood is approximately 0.830.
Note: It's important to note that the estimation process described above assumes that the data follows the given continuous PDF and that the maximum likelihood estimation provides the most likely value for the parameter based on the observed data.