Find the moment-generating function for a random variable W with density w-1 e 1 < w < 2, f (w) 0, е — else.
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!
![**Problem Statement:**
Find the moment-generating function for a random variable \( W \) with density:
\[
f(w) =
\begin{cases}
\frac{e^{w-1}}{e-1}, & 1 < w < 2, \\
0, & \text{else.}
\end{cases}
\]
**Explanation:**
This problem involves determining the moment-generating function (MGF) for a random variable \( W \) defined over a specific interval. The probability density function \( f(w) \) is given in a piecewise manner:
- For \( 1 < w < 2 \), the density is defined as \( \frac{e^{w-1}}{e-1} \).
- For all other values of \( w \), the density is 0.
The goal is to compute the MGF, which is a tool used in probability theory to find the moments of a random variable, given by the expectation \( M(t) = E[e^{tW}] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd25862a1-37f9-4b5c-af12-2fcd148d19b2%2Fb32e9062-cb2b-4940-9b89-16718f2e87d4%2F2lu3mtg_processed.png&w=3840&q=75)
Moment generating function is also known as mgf . It is a power series expansion .
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