1.9.18. Find the mean and the variance of the distribution that has the cdf x <0 0
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 1.9.18:**
Find the mean and the variance of the distribution that has the cumulative distribution function (cdf):
\[
F(x) =
\begin{cases}
0 & \text{for } x < 0 \\
\frac{x}{8} & \text{for } 0 \leq x < 2 \\
\frac{x}{2} - \frac{3}{16} & \text{for } 2 \leq x < 4 \\
1 & \text{for } 4 \leq x.
\end{cases}
\]
This piecewise function represents the cumulative distribution function (cdf) of a probability distribution. The cdf is defined in segments over different intervals of x:
- For \(x < 0\), the cdf is 0, indicating no probability mass below 0.
- For \(0 \leq x < 2\), the cdf is \(\frac{x}{8}\).
- For \(2 \leq x < 4\), the cdf is \(\frac{x}{2} - \frac{3}{16}\).
- For \(4 \leq x\), the cdf is 1, indicating that the total probability mass is accumulated by 4.
The problem asks for the mean and variance of this distribution. These are key statistical measures for understanding the central tendency and spread of the probability distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b0185de-0645-4c2a-aea4-e046d61ab5cb%2Fc0ce37c9-cab3-4363-9706-71578d5e235f%2F8d8sbn_processed.jpeg&w=3840&q=75)
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