A variable takes value x with frequency (n+x-1)Cx, x = 0, 1, 2, ...n. The mode of the variable is ..

Q: Why can't we just differentiate $f(x)$ to find the maximum?

That is a classic trap! Calculus is a powerful tool, but it is designed for continuous functions where variables can take any real value. Here, $x$ is restricted to integers $\{0, 1, 2, \dots, n\}$. We are dealing with a discrete sequence, not a smooth curve. We must use algebraic comparison instead to see how the sequence behaves step-by-step.

Q: Wait, if the ratio is $1 + \frac{n-1}{x}$, does that mean the frequency is always increasing?

Exactly! Since $n \ge 1$ and $x \ge 1$, the term $\frac{n-1}{x}$ is non-negative. This means the ratio is always $\ge 1$, implying $f(x) \ge f(x-1)$. The frequency never drops; it only climbs or stays flat as $x$ increases. This is the key to unlocking the problem.

Q: What if $n=1$? Does the logic still hold?

It absolutely does! If $n=1$, the ratio becomes $1 + \frac{1-1}{x} = 1$. This means $f(x) = f(x-1)$. The frequencies are equal, and the mode is still the largest value, $n=1$. The logic is robust even at the boundaries.

Question

A variable takes value $x$ with frequency $^{n+x-1}C_x$, $x = 0, 1, 2, \dots, n$. The mode of the variable is .........

Sigma Percentile · Accepted Answer

### Analyzing the Setup

In the study of discrete distributions, we often encounter variables where the frequency is defined by a combination formula. Given the frequency function $f(x) = \binom{n+x-1}{x}$, our goal is to identify the mode, which is the value of $x$ that maximizes this function.

Because $x$ is a discrete integer, we cannot rely on standard calculus derivatives. Instead, we must analyze the trend of the function by comparing the height of the current bar, $f(x)$, with the height of the bar immediately preceding it, $f(x-1)$.

### The Ratio Test

To determine if the function is growing, we examine the ratio $\frac{f(x)}{f(x-1)}$. This ratio acts as a growth factor: if it is greater than $1$, the function is increasing; if it is less than $1$, the function is decreasing.

We define the ratio as follows:

$$\frac{f(x)}{f(x-1)} = \frac{\binom{n+x-1}{x}}{\binom{n+x-2}{x-1}}$$

### The Algebraic Dance

Using the definition of combinations $\binom{N}{R} = \frac{N!}{R!(N-R)!}$, we expand the numerator and denominator. The numerator $\binom{n+x-1}{x}$ becomes:

$$\frac{(n+x-1)!}{x!(n-1)!}$$

The denominator $\binom{n+x-2}{x-1}$ becomes:

$$\frac{(n+x-2)!}{(x-1)!(n-1)!}$$

Performing the division by multiplying by the reciprocal, we observe significant cancellations:

$$\frac{f(x)}{f(x-1)} = \frac{(n+x-1)!}{x!(n-1)!} 	imes \frac{(x-1)!(n-1)!}{(n+x-2)!}$$

By simplifying the factorials using the property $k! = k 	imes (k-1)!$, we arrive at the simplified expression:

$$\frac{f(x)}{f(x-1)} = \frac{n+x-1}{x} = 1 + \frac{n-1}{x}$$

### Final Conclusion

We have derived the ratio $1 + \frac{n-1}{x}$. Given that $n \ge 1$ and $x \ge 1$, the term $\frac{n-1}{x}$ is always greater than or equal to zero.

This implies that the ratio is always $\ge 1$. Consequently, for every step from $x-1$ to $x$, the frequency $f(x)$ is non-decreasing.

Since the function is non-decreasing across its domain, the maximum value must occur at the upper boundary of the domain. Therefore, the mode of the variable is **$n$**.

Animated Solution for Mathematics - Statistics: A variable takes value $x$ with frequency $^{n + x - 1} C_{x}$ , $x = 0, 1, 2, \dots, n$ . The mode of the variable is .........

Visualized Solution

Understanding the Frequency Distribution

Strategy to Find the Maximum Frequency

Setting up the Ratio $\frac{f ( x )}{f ( x - 1 )}$

Expanding the Combinations

Simplifying the Ratio

Simplifying the Factorials

Final Ratio Expression

Analyzing the Growth of Frequency

Finding the Mode

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

Analyzing the Setup

The Ratio Test

The Algebraic Dance

Final Conclusion

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Animated Solution for Mathematics - Statistics: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A variable takes value x with frequency n+x−1Cx​, x=0,1,2,…,n. The mode of the variable is .........