Problem 1 A common way to study the effects of a treatment (e.g. a new medicine) is to run a randomized experiment, where some subjects are randomly assigned to a treatment, and others are randomly assigned to a control (e.g. a placebo). In order to estimate and conduct inference for these effects, researchers must consider the variance and covariance of random variables involved in the experiment. In this problem, we'll consider computing such variances and covariances. In this problem, we'll consider a randomized experiment with 20 subjects, where a random set of 10 subjects will be assigned to a treatment group, and the other 10 subjects will be assigned to a control group. Let T1, T2, ..., T20 denote the binary treatment indicator for each subject, where Ti = 1 if subject i is assigned to treatment, and Ti = 0 if assigned to control. Thus, each 20 Ti is said to follow a Bernoulli distribution with probability 100 = 1; one can find that, in this case, E[T]. This should be intuitive, because each of the 20 subjects has an equal chance of being in treatment or control. = 2 For this problem, answer the following two questions. Part A For this part, compute the following quantities: E[T}] Var(T1) E[T1 T2=1] ding Here, T1 and T2 denote the treatment indicator for the first and second subjects, respectively. Please be sure to show your work and explain how you arrived at your answer. Furthermore, for each of these quantities, please write your final answer as a single fraction. (Hint: Even though the subjects are assigned to treatment at random, T and T2 are still dependent-i.e., they are not independent. The reason is that we know we'll assign exactly 10 out of 20 subjects to treatment. Thus, another way to think about E[T|T2 = 1] is: What is the chance the first subject is assigned to treatment, given that we know the second subject is assigned to treatment?) Part B Cov(T1, T2) For this part, compute the following quantities: Cov (T1+T2, T2+T3) Here, T1 and T2 are defined the same as Part A, and T3 denotes the treatment indicator for the third subject. Please be sure to show your work and explain how you arrived at your answer. For each of these quantities, please write your final answer as a single number (which can be a fraction or a number with decimals).