Likelihood Function Homework: Data, Parameter & Solution

[hasan_researc]

Homework Statement

A series of independent measurements of the length of a rod delivers the following
result (units of cm):

{57.0, 49.3, 60.6, 74.5, 58.6, 62.4} (1)

Assume that the measurements are Gaussian distributed, with the standard deviation
of each measurement being 5.0 cm.

(a) What is the likelihood function for each measurement? Identify explicitly the data and the unknown parameter.

Homework Equations

No idea!

The Attempt at a Solution

No idea!

 

[mighty2000]

Hello,

Thank you for sharing your measurements with us. I would like to help you understand the likelihood function for each measurement.

The likelihood function is a function that represents the probability of obtaining a certain set of measurements given a specific unknown parameter. In this case, the unknown parameter is the true length of the rod.

To calculate the likelihood function for each measurement, we can use the Gaussian distribution formula:

f(x) = (1/σ√2π) * e^(-1/2((x-μ)/σ)^2)

Where:
f(x) is the probability density function
μ is the mean (in this case, the unknown parameter)
σ is the standard deviation (given as 5.0 cm)
x is the measurement value

Plugging in the values from your measurements, we get the following likelihood functions:

f(x1) = (1/5√2π) * e^(-1/2((57.0-μ)/5)^2)
f(x2) = (1/5√2π) * e^(-1/2((49.3-μ)/5)^2)
f(x3) = (1/5√2π) * e^(-1/2((60.6-μ)/5)^2)
f(x4) = (1/5√2π) * e^(-1/2((74.5-μ)/5)^2)
f(x5) = (1/5√2π) * e^(-1/2((58.6-μ)/5)^2)
f(x6) = (1/5√2π) * e^(-1/2((62.4-μ)/5)^2)

I hope this helps you understand the likelihood function for each measurement. If you have any further questions, please let me know.