Chapter 7 Flashcards
What do limit theorems describe?
The behavior of averages of independent and identically distributed (iid) random variables as the sample size grows.
What are the two fundamental limit theorems in probability?
The Law of Large Numbers (LLN) and the Central Limit Theorem (CLT).
What does the Law of Large Numbers (LLN) state?
The sample mean converges to the expected value as the sample size approaches infinity.
What does the Central Limit Theorem (CLT) state?
The distribution of the sample mean approaches a normal distribution as the sample size increases.
What is a random variable realization?
A specific observed value from a random experiment.
What is stochastic simulation?
Generating realizations of random variables using a computer.
What does the sample mean represent?
The average of a set of iid random variables.
What is the formula for the sample mean?
X̄n = (1/n) Σ Xi, where Xi are iid random variables.
What is the expectation of the sample mean?
E(X̄n) = µ, meaning the expected value remains the same as the population mean.
What happens to the variance of the sample mean as n increases?
Var(X̄n) = σ²/n, meaning variance decreases as sample size increases.
What is Chebyshev’s inequality?
P(|X - µ| ≥ kσ) ≤ 1/k², providing bounds on probability deviations.
How does Chebyshev’s inequality help prove the LLN?
It shows that the probability of large deviations from the mean approaches zero as n grows.
What is the practical implication of the LLN?
Relative frequencies of outcomes converge to their true probabilities over many trials.
How does the LLN affect histograms?
Histograms of large samples converge to the true probability distribution.
What is the standard normal variable in the CLT?
Zn = (X̄n - µ) / (σ/√n) approximately follows N(0,1) for large n.
Why is the CLT important?
It allows for approximating probabilities of sums and averages using the normal distribution.
How can the CLT be applied to sums of random variables?
The sum of iid variables follows a normal distribution for large n.
What is an example application of the CLT?
Estimating the probability of more than 120 errors in a data packet.
What is an example of using the CLT in decision-making?
Determining how many sandwiches to prepare for a party with 95% confidence.
What is the continuity correction in normal approximations?
Adjusting a discrete probability calculation by using P(X ≥ x) ≈ P(X ≥ x - 0.5).
Why is a continuity correction needed?
Because the normal approximation is continuous while the binomial distribution is discrete.
What is the standard normal distribution used for in the CLT?
To approximate probabilities of sample means and sums.
What happens to the distribution of sums of iid variables as n grows?
They approach a normal distribution regardless of the original distribution.
How does increasing n affect the accuracy of the normal approximation?
Larger n improves the approximation, making it more accurate.
What is an example of a CLT application in communication systems?
Modeling the number of errors in a data packet transmission.
What is a practical implication of the LLN in statistics?
Sample averages provide reliable estimates of population parameters.
What is the key assumption for the CLT to hold?
The variables must be iid with finite mean and variance.
How can the CLT be used for hypothesis testing?
It allows for approximating sampling distributions of test statistics.
What is an example of a poor CLT approximation?
Using a small n for a highly skewed distribution, leading to inaccurate normal approximation.
How does the CLT justify normal-based confidence intervals?
It ensures sample means follow a normal distribution for large enough n.
