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Chapter 23: Problem 7

If a variable \(x\) takes values \(0,1,2, \ldots, n\) with frequencies proportional to the binomial coefficients \({ }^{n} C_{0}\), \({ }^{n} C_{1},{ }^{n} C_{2}, \ldots,{ }^{n} C_{n}\), then the Var \((x)\) is (A) \(\frac{n^{2}-1}{12}\) (B) \(\frac{n}{2}\) (C) \(\frac{n}{4}\) (D) None of these

Short Answer

Expert verified
(C) \(\frac{n}{4}\)

Step by step solution

01

Understand the Setup

The problem states that a variable \( x \) takes values from \( 0 \) to \( n \) with frequencies proportional to binomial coefficients \( ^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_n \). This means that these values follow a binomial distribution with parameters \( n \) and \( p = \frac{1}{2} \). This is because each \( x \) value is chosen according to the probability of achieving \( x \) successes in \( n \) trials with success probability \( \frac{1}{2} \) (which defines a symmetric binomial distribution).
02

Recall the Variance Formula for Binomial Distribution

The variance of a binomial distribution \( \text{Var}(x) \) for a random variable \( x \) with parameters \( n \) (number of trials) and \( p \) (probability of success) is given by the formula \( \text{Var}(x) = np(1-p) \).
03

Apply to Given Parameters

Given that \( p = \frac{1}{2} \), substitute into the variance formula: \[ \text{Var}(x) = n\left(\frac{1}{2}\right) \left(1 - \frac{1}{2}\right) = n \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{n}{4}. \]
04

Identify the Correct Answer from Options

Compare the calculated variance \( \frac{n}{4} \) with the given options. Option (C) \( \frac{n}{4} \) matches our calculation.

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Key Concepts

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

variance of binomial distribution
The variance of a binomial distribution helps us understand how much variation or spread there is within a set of data. For a binomial distribution with parameters \( n \) (number of trials) and \( p \) (probability of success in each trial), the variance is determined by the formula \( \text{Var}(x) = np(1-p) \). This formula represents why variance is directly related to the parameters \( n \) and \( p \). The number of trials \( n \) dictates the amount of data, and the probability \( p \) impacts the expected outcomes. Hence, variance gives insight into the reliability and predictability of outcomes in a binomial experiment. To grasp these changes, consider
  • A larger \( n \), indicating more trials, typically increases the variance, suggesting more potential variation.
  • A \( p \) closer to 0 or 1 decreases the variation, as results lean heavily towards consistent success or failure.
Understanding the variance enables statisticians and scientists to predict the expected spread of outcomes in binomial distributions.
binomial coefficients
Binomial coefficients play a crucial role in algebra and probability and are typically expressed as \( { }^{n}C_{k} \) or simply \( C(n, k) \). These coefficients count the number ways of choosing \( k \) successes in \( n \) trials, providing the combinatorial backbone for the binomial distribution.The value of a binomial coefficient is calculated using the formula:\[ { }^{n}C_{k} = \frac{n!}{k!(n-k)!}. \]This formula relies on factorial calculations to determine the coefficients, which describe the number of different combinations possible in a given scenario. Key points to consider about binomial coefficients:
  • They are symmetric, meaning \( { }^{n}C_{k} = { }^{n}C_{n-k} \).
  • Each binomial coefficient corresponds to the entries of Pascal's Triangle.
Binomial coefficients are not only fundamental in probability but also appear in various fields such as algebra, statistics, and calculus.
probability distributions
Probability distributions provide us with a mathematical framework for understanding the likelihood of different outcomes in a random process. A probability distribution assigns a probability to each possible outcome of a statistical experiment.There are several types of probability distributions, and the binomial distribution is one such straightforward example. It describes the probability of obtaining a fixed number of successes in a specific number of independent trials, which can result in questions like those in the original exercise. Consider these features of binomial probability distributions:
  • The distribution is defined for trials with two possible outcomes: success or failure.
  • Each trial is independent, meaning the outcome of one trial does not affect another.
  • The distribution is controlled by two parameters, \( n \) (number of trials) and \( p \) (probability of success for each trial).
  • Binomial distribution is symmetric when \( p = 0.5 \), which reflects a fair set of trials.
By using probability distributions, we can accurately model real-world scenarios and make statistical inferences based on our findings.

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Most popular questions from this chapter

The G.M. of the number \(3,3^{2}, 3^{3}, \ldots, 3^{3 n}\) is (A) \(3^{\frac{n}{2}}\) (B) \(3^{\frac{3 n}{2}}\) (C) \(3^{\frac{3 n+1}{2}}\) (D) \(3^{\frac{n+1}{2}}\)

For two data sets, each with size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is (A) \(\frac{11}{2}\) (B) 6 (C) \(\frac{13}{2}\) (D) \(\frac{5}{2}\)

Let \(x_{1}, x 2, \ldots, x_{n}\) be \(n\) observations such that \(\sum x_{i}^{2}=400\) and \(\sum x_{i}=80 .\) Then a possible value of \(n\) among the following is [2005] (A) 15 (B) 18 (C) 9 (D) 12

The standard deviation of a distribution is 30 and each item is raised by 3 , then new S.D. is (A) 32 (B) 28 (C) 27 (D) None of these

If the variate of a distribution takes the values 1,2 , \(3, \ldots n\) with frequencies \(n, n-1, n-2, \ldots .3,2,1\), then mean value of the distribution is (A) \(\frac{n(n+2)}{3}\) (B) \(\frac{n(n+1)(n+2)}{6}\) (C) \(\frac{n+2}{3}\) (D) \(\frac{(n+1))(n+2)}{6}\)
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