This question will cover statistical inference, the relationship between a population parameter and the sample average.

Suppose you toss a coin and are interested in the probability of have a head. Assuming the coin is not rigged, the random variable X defined as 1 if it is a head and 0 if it is a tail is described by a Bernoulli distribution which takes the value 1 (head) with the population probability p = 0.5.

1. Plot the histogram of your sample
2. Estimate the sample average and the sample variance of X. Write down the formula first and then give the estimate. How does the sample average differ from the population average?
3. Generate another sample of N = 10 and repeat 3 and 4. Compare the histogram, and the sample average, sample variance. Explain.
4. Generate 100 samples of N = 10 observations and for each sample calculate the sample average. Then plot the sample averages (histogram/distribution of sample averages)