Question

8. For each i = 1, 2, ..., 10, Xi is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? [10]

Answer 1

Since the given coin is a fair coin, so we have

P(X_i = 0) = 1/2 = 0.5

P(X_i = 1) = 1/2 = 0.5

So E(X_i) = 0*0.5 + 1*0.5 = 0.5

Let X = X₁ + X₂ + ... + X₁₀

_{E(X) = E(X1) + E(X) + ... + EX 10) = 10 * 0.5 = 5}

The covariance terms is vanishes because the coins tossed independently.

Answer:

E(X) = 5

Var(X) = 2.5

b) Since X is the summation of 10 independent Bernoulli's trials with probability of successes 0.5, so the distribution of X is binomial with parameters n = 10 and p = 0.5

Answer: X follows binomial distribution with parameters n = 10 and p = 0.5