Patients are given a drug treatment and then evaluated. symptoms either improve, degrade or remain the same with probabilities 0.4, 0.1, 0.5, respectively. assume that four independent patients are treated and let x and y denote the number of patients who improve or degrade. are x and y independent? calculate the covariance and the correlation between x and y.

The data represents the possible probabilities of treatment levels of a drug. From the given information, the patient can improve, degrade and remain the same with probabilities 0.4, 0.1 and 0.5 respectively. Let X denotes the number of patients who can improve in the treatment. Let Y denotes the number of patients who can degrade in the treatment. The two random variables X and Y are dependent, because Y depends upon the remaining cases after the random variable X noted.

Calculate the covariance and correlation for the given data by using the given information. Use the formula below to calculate the covariance and correlation.

Covariance:

Calculate the joint probability distribution function for 4 randomly selected students. The number of patients who can improve in the treatment is 0 and the number of patients who can degrade in the treatment is 0. It means that the patients remain in the same condition. Thus, the probability of remains same condition is 0.5, then the joint probability is P(X=0 ∩ Y=0) = 0.5^4 = 0.0625.

The number of patients who can improve in the treatment is 1 and the number of patients who can degrade in the treatment is 0. It means that the rest of 3 patients remains in the same condition. The joint probability is P(X=1 ∩ Y=0) = (4 1 )*0.4*0.5^3 = 0.2

Calculating the covariance and correlation we get E(XY) = 0.48. E(X) = 1.6. E(Y) = 0.4.