Inferential statistics-maximum likelihood function

[kimkibun]

Homework Statement

suppose that n cylindrical shafts are selected at random from the production of the machine and their diameters and lengths are measured. it is found that N₁₁ have both measurements within the tolerance limits, N₁₂ have satisfactory lengths but unsatisfactory diameters, N₂₁ have satisfactory diameters but unsatisfactory lengths, and N₂₂ are unsatisfactory to both measurements. ƩN_ij=n. each shaft may be regarded as a drawing from a multinomial population with density

p₁₁^x₁₁p₁₂^x₁₂p₂₁^x₂₁(1-p₁₁-p₁₂-p₂₁)^x₂₂; for x_ij=0,1; Ʃp_ij=1​

having three parameters. what are the maximum likelihood estimates of the parameter if N₁₁=90, N₁₂=6, N₂₁=3, and N₂₂=1?

Homework Equations

The Attempt at a Solution

my solution is attached. i just want to know if my answers are correct.

 

[mighty2000]

Hello, thank you for sharing your solution. Your answers seem to be correct based on the information provided in the forum post. However, it would be helpful to see your calculations and steps in finding the maximum likelihood estimates so that we can provide feedback and ensure that your approach is correct. Additionally, it would be beneficial to include any assumptions or limitations that you made in your solution. Overall, your solution appears to be reasonable and well-supported.