#StackBounty: #probability #bayesian #mathematical-statistics #posterior #conjugate-prior Help with the prior distribution

Bounty: 50

The question is as follows:

Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg.
The value of the spring stiffness is unknow and based on the experience and judgement the following is assumed. Value of stiffness is in the following range [0.5, 1.5] N/m.

To have a more accurate estimate of the value of the stiffness an experiment is performed where in the natural frequency of the system is observed. The following observation are made:

  Observation 1     Freq = 1.021 rad/sec
  Observation 2     Freq = 1.015 rad/sec
  Observation 3     Freq = 0.994 rad/sec
  Observation 4     Freq = 1.005 rad/sec
  Observation 5     Freq = 0.989 rad/sec
  1. Based on the information provided write the functional form of prior PDF.
  2. Plot the likelihood function with different number of observations.
  3. Based on the information provided write the functional form of the posterior PDF.
  4. Plot the posterior distribution.

My work so far:

spring constant $$k = sqrt{{w}/{m}}$$
m = 1kg, so $$w = k^{2}$$.

$$k sim Uniform(0.5, 1.5)$$,

so pdf of w = $$ f(w) = 2w$$

where $$w epsilon [sqrt{0.5},sqrt{1.5}] $$

So prior distribution is linear in the range root(0.5), root(1.5).

$$Likelihood = L = 2^{5}(1.0211.015..0.989) approx 2.04772 $$

This is what I have done so far. I am new to Bayesian inference and I am not sure how to proceed after this or if what I have done so far is correct. Pleas advice on how to find the posterior function.


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