5.10  Review Exercises

5.1. Using the definition of conditional probability, prove the Bayes’ rule.
5.2. Bayes rule. A person is diagnosed with “positive” for a mortal disease. However, the diagnostic is correct only with the following probabilities: in 90% of the cases when the disease is present, i. e., P(diagnostic= “positive” / disease = “true”)=0.9 and 80% when there is no disease: P(diagnostic= “negative” / disease = “false”)=0.8. Hence, the false-negative probability is 10% and the false-positive is 20%. Suppose the disease appears in 2 for each group of 100 people. What is the probability of actually having the disease given a positive diagostic?
5.3. MLE for Gaussian parameters. Assume $N$ independent and identically-distributed (iid) examples $x[n],n\; =\; 1,\dots ,N$ should be fitted to a Gaussian distribution. Prove that the maximum likelihood estimation (MLE) for the mean $\mu$ is $\widehat{\mu }=\frac{1}{N}{\sum }_{n=1}^{N}x[n]$. You do not need to calculate, but check the steps for proving the MLE for the variance ${\widehat{\sigma }}^2=\frac{1}{N}{\sum }_{n=1}^N{(x[n]\; -\; \mu )}^2.$