If we look at college course enrollment it is known that 28% of college students take at least one mathematics or statistics course in a given semester. Suppose we take a random sample of 13 college students in Fall 2019. Let X = the number of these students who are taking at least one mathematics or statistics course. d) What is the expected value of X? e) What is the variance of X?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
d) What is the
e) What is the variance of X?
Solution:
d.
Let X be the number of these students who are taking at least one mathematics or statistics course and n be the sample of college students.
From the given information, probability that college student take at least one mathematics or statistics course is 0.28 and n=13.
Here, the college students are independent and probability of success is constant. Hence, X follows binomial distribution with parameters n=13 and p=0.28.
The probability mass function of binomial random variable X is
Then, the expected value of X is
Thus, the expected value of X is 3.64.
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