Let μ be the mean and σ be the standard deviation of the distribution Xi: 0, 1, 2, 3, 4, 5; fi: k+2,

Q: Wait, why do we reject the negative root for $k$?

In statistics, frequencies represent the number of times an event occurs. A count cannot be negative. If you look at the frequency $f_5 = k - 3$, a negative $k$ would result in a negative frequency, which is physically impossible. Always check your domain constraints!

Q: Why can't we just use energy conservation? (Wait, this is stats, not physics).

I see you're thinking like a physicist! While we don't have energy conservation here, we have the 'conservation of moments'. The identity $\sigma^2 = E[X^2] - \mu^2$ is a powerful tool that saves us from calculating the mean $\mu$ explicitly, reducing the chance of arithmetic errors.

Q: Is there a faster way to calculate $\mu^2 + \sigma^2$?

Yes! As we discussed, $\mu^2 + \sigma^2$ is exactly the second moment $E[X^2]$. By recognizing this, we bypass the need to find $\mu$ and then square it, which simplifies the algebra significantly.

Question

Let $\mu$ be the mean and $\sigma$ be the standard deviation of the distribution $$\begin{array}{|l|l|l|l|l|l|l|} \hline X_i & 0 & 1 & 2 & 3 & 4 & 5 \ \hline f_i & k + 2 & 2k & k^2 - 1 & k^2 - 1 & k^2 + 1 & k - 3 \ \hline \end{array}$$ where $\sum f_i = 62$. If $[x]$ denotes the greatest integer $\le x$, then $[\mu^2 + \sigma^2]$ is equal to

Options:

(A) 9

(B) 8

(C) 7

(D) 6

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the value of $[\mu^2 + \sigma^2]$ for a given frequency distribution where $X_i \in \{0, 1, 2, 3, 4, 5\}$ and frequencies $f_i$ depend on a parameter $k$. The total frequency is given as $\sum f_i = 62$.

To find $k$, we sum the given frequencies:
$$(k+2) + 2k + (k^2-1) + (k^2-1) + (k^2+1) + (k-3) = 62$$

Simplifying this expression leads to the quadratic equation:
$$3k^2 + 4k - 64 = 0$$

### Solving for the Parameter

Solving the quadratic equation $3k^2 + 4k - 64 = 0$ yields two potential roots:
$$k = 4 \quad 	ext{and} \quad k = -\frac{16}{3}$$

In statistical distributions, frequencies must be non-negative. Checking the frequency $f_5 = k-3$, we observe that $k = -\frac{16}{3}$ results in a negative frequency, which is physically impossible. Therefore, we reject the negative root and accept **$k = 4$**.

### The Elegant Shortcut

We are asked to compute $[\mu^2 + \sigma^2]$. Rather than calculating the mean $\mu$ and variance $\sigma^2$ separately, we utilize the fundamental relationship:
$$\sigma^2 = E[X^2] - \mu^2$$

Rearranging this identity, we find:
$$\mu^2 + \sigma^2 = E[X^2]$$

This simplifies our task significantly, as we only need to calculate the second moment of the distribution, defined as:
$$E[X^2] = \frac{\sum f_i X_i^2}{\sum f_i}$$

### Final Calculation

With $k=4$, the frequencies are determined, and the sum of $f_i X_i^2$ is calculated as:
$$\sum f_i X_i^2 = 29k^2 + 27k - 72$$

Substituting $k=4$ into this expression:
$$29(16) + 27(4) - 72 = 464 + 108 - 72 = 500$$

Finally, we divide by the total frequency $\sum f_i = 62$:
$$E[X^2] = \frac{500}{62} \approx 8.06$$

Applying the greatest integer function, we arrive at the final result:
$$[8.06] = \mathbf{8}$$

Select Answer:

Visualized Solution

Understanding the Objective

Summing the Frequencies

Solving the Quadratic Equation

Determining the Valid Value of $k$

Relating Mean and Variance

Setting up Sum of Squares

Substituting the value of $k$

Final Calculation

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

Solving for the Parameter

The Elegant Shortcut

Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

Understanding the Objective

Summing the Frequencies

Solving the Quadratic Equation

Determining the Valid Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

Relating Mean and Variance

Setting up Sum of Squares

Substituting the value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

Final Calculation

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

Solving for the Parameter

The Elegant Shortcut

Final Calculation

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