Machine Learning Maximum Likelyhood Estimate Question (MLE) 13. In this problem

Question

Machine Learning Maximum Likelyhood Estimate Question (MLE)

13. In this problem you will use the prior and MLE estimates of a univariate Gaussian distribution to assign probability of a new instance to be in one of two classes (each class modeled as a Gaussian distribution). You will need to use the discussion in class about MLE for Gaussian distribution. You can also use the notebook to compute the exact values for the probability density. Please note that the norm function takes the standard deviation and not the variance) as a parameter Consider the example of analyzing weights (at-birth) of human babies (female) for different cities. WHO estimates state that the weights of girls born in Mexico City are normally distributed with mean weight of 4.4 lbs and standard deviation of 0.5 lbs. The weights of girls born in Boston are also normally distributed with mean weight o 5 lbs and standard deviation of 0.7 lbs. A recent sample from hospitals in both cities is shown below. Each ro corresponds to weights of 10 girls born in each city in the last 1 day. Mexico City 4.72, 4.15, 4.46, 4.23, 5.28, 4.07, 4.67, 4.87, 4.54, 3.75 Boston 5.76. 5.12, 5.64, 5.59, 5.54, 5.09, 5.46, 5.87. 5.71, 5.56 Given the prior WHO estimates and the new data, your task is to infer th city of birth for two newborn girls born with following weights: Baby A 4.6 lbs and Baby B 4.9 lbs. Hint: You need to compare the probability density for each baby's weight under two distributions for each city (class (1) Using the prior WHO estimates, and (2) Using the MLE estimates using the observed data. a) None of the two babies are more likely to be born in Mexico city by using the new data. b According to the new data, both babies are more likely to be have been born in Boston. Both babies are more likely to be born in Mexico city by WHO estimates. c) d) Baby A is more likely to be born in Mexico city by both WHO estimates and only using the new data.

Explanation / Answer

A normal distribution, sometimes called the bell curve, is a distribution that occurs naturally in many situations. For example, the bell curve is seen in tests like the SAT and GRE. The bulk of students will score the average (C), while smaller numbers of students will score a B or D. An even smaller percentage of students score an F or an A. This creates a distribution that resembles a bell (hence the nickname). The bell curve is symmetrical. Half of the data will fall to the left of the mean; half will fall to the right.
Many groups follow this type of pattern. That’s why it’s widely used in business, statistics and in government bodies like the FDA:

The empirical rule tells you what percentage of your data falls within a certain number of standard deviations from the mean:
• 68% of the data falls within one standard deviation of the mean.
• 95% of the data falls within two standard deviations of the mean.
• 99.7% of the data falls within three standard deviations of the mean.

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