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Chapter 4: Problem 20

Find the standard deviation. $$\begin{array}{|c|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} & {(x-\mu)^{2} P(x)} \\ \hline 2 & {0.1} & {2(0.1)=0.2} & {(2-5.4)^{2}(0.1)=1.156} \\\ \hline 4 & {0.3} & {4(0.3)=1.2} & {(4-5.4)^{2}(0.3)=0.588} \\ \hline 6 & {0.4} & {6(0.4)=2.4} & {(6-5.4)^{2}(0.4)=0.144} \\ \hline 8 & {0.2} & {8(0.2)=1.6} & {(8-5.4)^{2}(0.2)=1.352} \\ \hline \end{array}$$

Short Answer

Expert verified
The standard deviation is approximately 1.8.

Step by step solution

01

Calculate the Expected Value (Mean)

The expected value or mean \( \mu \) of a set of data is calculated as the sum of each value multiplied by its probability: \( \mu = \sum x \times P(x) \). For the given data, \( \mu = 0.2 + 1.2 + 2.4 + 1.6 = 5.4 \).
02

Calculate the Variance

The variance \( \sigma^2 \) is calculated by taking the sum of the squared differences from the mean, each multiplied by their respective probability. This is given as \( \sigma^2 = \sum (x - \mu)^2 \times P(x) \). Using the table, we compute: \( 1.156 + 0.588 + 0.144 + 1.352 = 3.24 \).
03

Calculate the Standard Deviation

The standard deviation \( \sigma \) is the square root of the variance, calculated in Step 2. So, \( \sigma = \sqrt{3.24} \approx 1.8 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Expected Value
The expected value, often referred to as the "mean," is a key concept in probability and statistics. It represents the average outcome one would expect from a probability distribution if they could repeat an experiment an infinite number of times.
The formula to find the expected value \( \mu \) is given by:
  • \( \mu = \sum x \times P(x) \)
This means you multiply each possible outcome \( x \) by its probability \( P(x) \) and sum up all these products.
For example, with the provided data, the calculation for the expected value became:
  • \( 0.2 + 1.2 + 2.4 + 1.6 = 5.4 \)
Thus, the expected value, or the average, is 5.4.
The Role of Variance
Variance measures how much the values in a set differ from the mean. It tells us whether the numbers are generally close to the mean or far away from it.
To calculate variance \( \sigma^2 \), we use the formula:
  • \( \sigma^2 = \sum (x - \mu)^2 \times P(x) \)
This involves subtracting the mean \( \mu \) from each outcome \( x \), squaring the result, multiplying by the probability \( P(x) \) and then adding these results together.
For instance, in our example, the variance was:
  • \( 1.156 + 0.588 + 0.144 + 1.352 = 3.24 \)
Unpacking Probability Distribution
A probability distribution assigns a probability to each possible outcome of a random event. It lays out how probabilities are spread over different outcomes.
In discrete probability distributions, like the exercise's table, each outcome \( x \) has a corresponding probability \( P(x) \), such that:
  • The sum of all probabilities \( \sum P(x) \) is 1.
This implies every possible outcome is considered, ensuring a complete picture of the event.
For example, in our problem, probabilities like 0.1 and 0.4 show how likely each value of \( x \) will occur.
Applying a Step-by-Step Solution
Breaking down problems into a sequence of simpler steps is a great way to confirm you understand each piece. Here's a recap of the process for our exercise:
**Step 1: Calculate the Expected Value (Mean)**
The expected value was determined by multiplying each outcome by its probability and summing all products, resulting in:\( \mu = 5.4 \).
**Step 2: Compute the Variance**
  • Using the expected value, the variance was calculated by assessing squared deviations, adjusted by probability, summed up to \( 3.24 \).
**Step 3: Find the Standard Deviation**
  • Finally, the standard deviation was found as the square root of the variance, approximately \( \sigma = 1.8 \).
Ensuring you understand each step builds confidence in tackling more complex distributions.

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Most popular questions from this chapter

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