1. Assume that Z is a standard normal random variable (i.e., it is a normal random variable with mean 0 and variance 1). Show that the moment generating function for Z is mz(t) = et/2. Recall that by definition, the moment generating function for Z is E[etZ]. 2. Let L = ez be a log-normal random variable. Use the moment generating function for Z to derive the mean and variance of L.
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!
![A distribution that comes up often is the log-normal distribution. Specifically, if X is a normal random
variable, then a random variable Y defined as e* is said to have a log-normal distribution. In this problem,
we will explore some properties of the log-normal distribution.
1. Assume that Z is a standard normal random variable (i.e., it is a normal random variable with mean
O and variance 1). Show that the moment generating function for Z is mz(t) = e²/2. Recall that by
definition, the moment generating function for Z is E[et].
2. Let L = e
= e² be a log-normal random variable. Use the moment generating function for Z to derive the
mean and variance of L.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6038fcf6-1d64-45aa-9476-97e6a64f1542%2F8685eb5c-8c13-4fd0-8ac6-75842048c6bd%2Ff1r2hyp_processed.png&w=3840&q=75)
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