Hi, i am fairly new on these forums, so please let me know, if information or formulation is lacking.
My question is related to the picture i have linked. My problem is i really don't understand how the likelihoods of $\theta$ is derived and my intuition of likelihood functions is lacking a lot. Could anyone explain to me how i would go on about solving these problems? I find (c), (d), (e) especially hard to grasp.
likelihood is actually the joint density function but the interpretation is a little different. Consider a sample from any distribution whose parameter(s) are unknown. For simplicity let's say Normal(a,1) i.e, mean =a ; variance =1. Now the maximum likelihood estimate of a means that for what value of a are you most likely to draw the sample that you have. Meaning that after you estimate a, the distribution is identified so you can always draw a random sample. The probability that you draw the sample you have is maximised with the mle. Now it's obvious that you will actually never get exactly the sample you have, since it's a continuous distribution so the probability of the random variable taking a particular value is always 0. Here you are maximising the Probability of drawing the sample.