Question

John and Jack found a coin on the sidewalk. They argued about the fairness of the coin. John claimed 40% to have Heads according to his careful observation of the coin. Jack doubted and in order to infer the fairness of the coin, he tossed the coin for 50 times and got the results as shown below with 1 representing as heads and 0 as tails:

## [1] 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

## [36] 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0

Let p∗ as the true probability to have heads for the coin. Note that p∗ is a characteristic of the coin, and we want to make some inference about this unknown parameter. And denote X as the random variable which takes 0 if tails show up or 1 if heads show up for tossing the coin.

1. What is the distribution of the random variable X?

2. Calculate the sample proportion pˆ∗ which is the proportion of “1”s shown in the above given sample. (Note that this hat version is your ”best” guess for the unknown p∗ based on the sample. These two notations, p∗ and pˆ∗, are indeed different.)

Answer 1