Find the standard deviation for the given probability distribution. P(x) 0.15 1 0.17 0.11 0.33 4 0.24 O 2.72 1.94 O 1.39 O 1.45 2.
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!
![### Finding the Standard Deviation for a Given Probability Distribution
#### Problem Statement
Determine the standard deviation for the provided probability distribution.
#### Probability Distribution Table
The table below presents the values of \(x\) and their corresponding probabilities \(P(x)\):
| x | P(x) |
|---|------|
| 0 | 0.15 |
| 1 | 0.17 |
| 2 | 0.11 |
| 3 | 0.33 |
| 4 | 0.24 |
#### Choices for Standard Deviation
Select the correct standard deviation value from the following options:
- 2.72
- 1.94
- 1.39
- 1.45
#### Explanation
To determine the correct answer, you must calculate the standard deviation of the given probability distribution. Here's the general approach:
1. **Calculate the Mean (μ)**:
\[
\mu = \sum [x \cdot P(x)]
\]
2. **Calculate the Variance (σ²)**:
\[
\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]
\]
3. **Calculate the Standard Deviation (σ)**:
\[
\sigma = \sqrt{\sigma^2}
\]
By applying these steps to the given table, you can find the correct standard deviation value. Select the value closest to your computed result from the provided options.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5f063df-177a-4dd3-b530-509ae96cf879%2F94226168-2ba0-47ef-9582-78959f0777c7%2F91e2fei_processed.png&w=3840&q=75)
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