Statistics: Probabilities and Central Limit Theorem

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Questions and Answers

What is the relationship between the mean and variance of a random sample drawn from a continuous distribution?

The variance reflects the spread around the mean. (correct)
The variance increases with an increase in the mean.
The mean is always greater than the variance.
The mean and variance are independent of each other.

In the context of independent identically distributed (i.i.d.) random variables, which statement is true?

Their mean and variance can vary.
Each variable is drawn from a different distribution.
They are dependent on each other.
They all share the same probability distribution. (correct)

What does the Central Limit Theorem (CLT) state about the distribution of sample means?

Sample means can never be normally distributed.
The distribution of sample means is always skewed.
Sample means are normally distributed for large sample sizes. (correct)
Sample means have a uniform distribution.

What is a moment generating function primarily used for in probability theory?

To characterize the entire distribution of a random variable. (B) Signup and view all the answers

What is the mean resistance (E(R)) calculated in the content?

1650 (A) Signup and view all the answers

How does the standard normal distribution differ from other normal distributions?

It has a mean of 0 and a variance of 1. (A) Signup and view all the answers

What does the simulation of R suggest about its distribution?

It is approximately normally distributed. (C) Signup and view all the answers

What theorem states that the sum or average of iid normal random variables is also normally distributed?

Central Limit Theorem (D) Signup and view all the answers

Why are R-values near the theoretical maximum (1732.5) considered unlikely?

All five resistances must exert high resistances. (D) Signup and view all the answers

What does the moment generating function (mgf) of R indicate?

It is not recognizable as coming from any known family of distributions. (C) Signup and view all the answers

In the context of the content provided, what does 'iid' stand for?

Independent and Identically Distributed (B) Signup and view all the answers

What will summing independent uniform variates likely produce in terms of distribution shape?

Bell-shaped distribution (C) Signup and view all the answers

What is the significance of R being more likely to be found 'centrally' located rather than at extremes?

It shows normal distribution behavior. (B) Signup and view all the answers

What is the relationship between the random variable X and the trials in a binomial experiment?

X is the number of successes in n trials. (D) Signup and view all the answers

Which statement correctly describes the distribution of the sample proportion $P^$ as the number of trials n increases?

$P^$ approaches a normal distribution if np and n(1-p) are both greater than or equal to 10. (B) Signup and view all the answers

What does the Central Limit Theorem (CLT) imply for sufficiently large n in a binomial distribution?

The sample mean and sample proportion are approximately normal. (C) Signup and view all the answers

What is the variance of the sample proportion $P^$ given that p is the probability of success?

$V(P^) = p(1 - p)/n$. (D) Signup and view all the answers

How are the random variables $X_i$ defined in a binomial experiment?

As the indicator of success for each trial. (B) Signup and view all the answers

What characteristic defines the random variables $X_1, X_2, ..., X_n$ in the context of binomial trials?

They are identically distributed and independent. (C) Signup and view all the answers

What is the expected value of the sample proportion $P^$?

p (D) Signup and view all the answers

Which condition must be satisfied for the approximation of the distribution of sample proportions $P^$ to be reliable?

Both np and n(1-p) should be greater than or equal to 10. (C) Signup and view all the answers

Study Notes

Calculating Probabilities and Expected Values

Calculating the probabilities and expected values of statistics can be tedious, especially for larger sample sizes and more complex distributions.
There are general relationships between the expected value and variance of a statistic, and the mean and variance of the original population.

The Central Limit Theorem

The Central Limit Theorem (CLT) states that, under certain conditions, the distribution of the sample mean approaches a normal distribution as the sample size increases.
This holds true even if the original population distribution is not normal.
The CLT is a powerful tool because it allows us to approximate the distribution of many statistics, even when the underlying distribution is unknown.
The approximation is good when np ≥ 10 and n(1-p) ≥ 10, where n is the sample size and p is the probability of success in a Bernoulli trial.
The sample size required for a good approximation depends on the value of p. When p is close to 0.5, the distribution of each Bernoulli variable is reasonably symmetric. However, the distribution is skewed when p is near 0 or 1.

Example of a Simulated Distribution

Simulated results can provide insights into the distribution of a statistic, especially when the exact distribution is difficult to calculate.
In the provided example, the resistance of a system is simulated by summing five independent uniformly distributed random variables.
The simulation suggests that the distribution of the system’s resistance is approximately normal, even though the distribution of each individual resistance is uniform.
This demonstrates that the CLT can apply even when the underlying distribution is not normal, indicating that the sum or average of a large number of independent random variables will tend to be normally distributed.

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Description

Explore the concepts of probabilities, expected values, and the Central Limit Theorem in this quiz. Understand how sample sizes affect distribution approximation and the relationships between statistical variables. Test your knowledge on these fundamental statistical principles.