A psychiatrist believes that 55% of all people who visit doctors have problems that are of a psychosomatic nature. She selects 25 patients at random to test her theory. Let Xbe the number of patients with psychosomatic problems out of the 25 patients.

a) What is the sample space for X?

b) What probability distribution can you use to model X? Make sure to name the distribution and the values of any parameters.

c) Assuming the psychiatrist's theory is true, what is the expected value of X, the number of the 25 patients who have psychosomatic problems?

d) What is the standard deviation of X, assuming the theory is true?

e) What is the exact probability that X < 4 in any given case? Write down the formula you would use with numbers substituted but use R studio to calculate the answer.