Suppose we toss a coin 3 times. We denote head as H and tail as T....

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Suppose we toss a coin 3 times. We denote head as H and tail as "T". We assume the probability of tail is P(T) 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by for every possible value k. Check that P(X = k) P(X = P(X = k) = 1. 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X. 4. Do the followings with R: a. Create a plot of the pf of X. b. Create a plot of the cdf of X. c. Draw a random sample of size 1000 from Binomial(n = 3, p = 0.3). Then plot a histogram and calculate the sample mean and the sample variance. Comment briefly on these outputs compared to the results from question 1-3. Suppose we toss a coin 3 times. We denote head as H and tail as "T". We assume the probability of tail is P(T) 0.7. Let X be the number of "H"s. 1. What are the possible values of X? Find the probability function(pf) of X, which is defined by for every possible value k. Check that P(X = k) P(X = P(X = k) = 1. 2. Find the cumulative distribution function (cdf) of X. 3. Calculate the expectation and the variance of X. 4. Do the followings with R: a. Create a plot of the pf of X. b. Create a plot of the cdf of X. c. Draw a random sample of size 1000 from Binomial(n = 3, p = 0.3). Then plot a histogram and calculate the sample mean and the sample variance. Comment briefly on these outputs compared to the results from question 1-3.