Question

Let Xi be a random variable that takes on the value 1 with probability p and the value 0 with probability q = 1 − p. As we have learnt, this type of random variable is referred to as a Bernoulli trial. This is a special case of a Binomial random variable with n = 1.

a. Show the expected value that E(Xi)=p, and Var(Xi)=pq

b. One of the most common laboratory tests performed on any routine medical examination is a blood count. The two main aspects of a blood count are (1) counting the number of white blood cells (the “white count”) and (2) differentiating the white blood cells that do exist into five categories—namely, neutrophils, lymphocytes, monocytes, eosinophils, and basophils (called the “differential”). Both the white count and the differential are used extensively in making clinical diagnoses. We concentrate here on the differential, particularly on the distribution of the number of neutrophils k out of 100 white blood cells (which is the typical number counted). We will see that the number of neutrophils follows a binomial distribution. In this case, let X1, . . . , Xn indicate the neutrophils status among 100 white blood cells: n = 100 and Xi = 1 if the ith white blood cell is a neutrophil and Xi = 0 if the ith white blood cell is not a neutrophil, where i = 1, . . . , 100. Show the expected value that E(Xi) and Var(Xi).

c. Can we approximate the distribution of X by a normal distribution? Please explain why or why not. If we can, what is the pdf of the normal distribution to use.

Answer 1