Question

Let \(x\) be a binary explanatory variable and suppose \(P(x=1)=\rho\) for \(0<\rho<1\). i. If you draw a random sample of size \(n\), find the probability-call it \(\gamma_{n}-\) that Assumption \(\mathrm{SLR} .3\) fails. [Hint: Find the probability of observing all zeros or all ones for the \(x_{i} .\) ] Argue that \(\gamma_{n} \rightarrow 0\) as \(n \rightarrow \infty\). ii. If \(\rho=0.5,\) compute the probablity in part (i) for \(n=10\) and \(n=100 .\) Discuss. iii. Do the calculations from part (ii) with \(\rho=0.9 .\) How do your answers compare with part (ii)?

Short answer

As \( n \to \infty \), \( \gamma_n \to 0 \). For \( n = 10, \gamma_n \approx 0.00195\) with \( \rho = 0.5\), while \( \gamma_n \approx 0.34868\) with \( \rho = 0.9\).

Step-by-step solution

Understanding the Sample Space

Consider a binary explanatory variable \( x \) with \( P(x=1)=\rho \). Any draw for \( x \) can result in \( x=0 \) or \( x=1 \). When drawing \( n \) samples, the probability of drawing all zeros or all ones needs to be cleared.

Calculate Probability of All Zeros or All Ones

The probability that all \( n \) samples are zeros is \((1-\rho)^n\). The probability that all \( n \) samples are ones is \(\rho^n\). Therefore, the probability \( \gamma_n \) that SLVR.3 fails, i.e., only one value is observed in the sample, is the sum: \[ \gamma_n = (1-\rho)^n + \rho^n. \]

Asymptotic Behavior of Gamma

To argue that \( \gamma_n \rightarrow 0 \) as \( n \rightarrow \infty \), observe that both \( \rho^n \) and \( (1-\rho)^n \) tend towards zero at an exponential rate. Hence, \( \gamma_n \rightarrow 0 \).

Compute \( \gamma_{10} \) for \( \rho = 0.5 \)

Substitute \( \rho = 0.5 \) into the formula: \( \gamma_{10} = (0.5)^{10} + (0.5)^{10} = 2 \cdot (0.5)^{10} = \frac{2}{1024} = \frac{1}{512} \approx 0.00195. \)

Compute \( \gamma_{100} \) for \( \rho = 0.5 \)

Using the same formula, \( \gamma_{100} = (0.5)^{100} + (0.5)^{100} = 2 \cdot (0.5)^{100}. \) Knowing that \( (0.5)^{100} \) is extremely small, makes \( \gamma_{100} \) almost zero.

Compute \( \gamma_{10} \) for \( \rho = 0.9 \)

Substitute \( \rho = 0.9 \) into the formula: \[ \gamma_{10} = (0.1)^{10} + (0.9)^{10} = 10^{-10} + 0.34867 \approx 0.34868. \]

Compute \( \gamma_{100} \) for \( \rho = 0.9 \)

Now, compute \( \gamma_{100} = (0.1)^{100} + (0.9)^{100} \). With \( (0.9)^{100} \) also being tiny, the total \( \gamma_{100} \) approaches zero, but is much larger if \( \rho = 0.9 \) than if \( \rho = 0.5 \).

Discussion

When \( \rho = 0.5 \), extreme values are much less likely compared to when \( \rho = 0.9 \). The exponential decay in both scenarios shows dependence on \( \rho \). For \( n \) large and \( \rho \) closer to 0.5, \( \gamma_n \) is nearly zero.

Key concepts

Binary Variable

A binary variable is a fundamental concept in statistics where the variable can take on only two possible values. These values are often represented as 0 and 1. For example, in a medical study, a binary variable might indicate whether a patient has a disease (1) or does not have a disease (0). This type of variable is crucial in many fields, including economics, social sciences, and biostatistics.

• In the given exercise, the binary variable is denoted as \(x\), where \( P(x=1)=\rho \) and \( P(x=0)=1-\rho \).
• The probability \( \rho \) is a parameter that determines the likelihood of \(x\) being 1 in any single draw.
• Binary variables are used to model experiments and understanding these probabilities forms the basis of statistical inference for binary data.

Understanding how binary variables operate is key to grasping more complex statistical concepts and making probability calculations.

Asymptotic Behavior

Asymptotic behavior in statistics refers to the behavior of a statistic as the sample size \( n \) becomes very large. This concept is vital for understanding how probabilities and estimators behave in large samples.

• In the problem, \( \gamma_n \), the probability that SLVR.3 fails, is shown to approach zero as \( n \rightarrow \infty \).
• This happens because both probabilities \( \rho^n \) and \((1-\rho)^n\) decay exponentially. As \( n \) increases, these tiny numbers make \( \gamma_n \) shrink towards zero.
• Asymptotic behavior helps statisticians predict and verify the consistency and reliability of estimators in large data sets.

Analyzing the asymptotic properties of a probability can reveal much about the long-term stability and reliability of statistical models.

Probability Calculations

Probability calculations involve determining the likelihood of certain outcomes. In the context of binary variables, it’s about finding the probability of drawing either all ones or all zeros in a sample.

• The probability of observing all zeros in a random sample is \((1-\rho)^n\).
• The probability of observing all ones is \(\rho^n\).
• Total probability \(\gamma_n\) of SLVR.3 failing is a sum of these probabilities: \[(1-\rho)^n + \rho^n.\]

These calculations are fundamental to understanding data behavior and are widely used in hypothesis testing and decision-making processes in statistics.

Sample Size

Sample size, denoted as \( n \), is a critical factor in statistical studies as it affects the reliability and accuracy of the inferences drawn from data.

• A larger sample size generally leads to more reliable estimates, as seen in the decay of \( \gamma_n \) as \( n \) increases.
• In the exercise, we see how \( n=10 \) and \( n=100 \) yield different probabilities under different values of \( \rho \).
• For \( \rho = 0.5 \), \( \gamma_{10} \) is very small, indicating a low probability of extreme outcomes, whereas \( n = 100 \) makes it almost zero.
• Conversely, for \( \rho = 0.9 \), the probabilities are more significant with smaller sample sizes, but diminish with larger \( n \).

Choosing an adequate sample size is essential to ensure the statistical results are valid and representative of the true population.