Problem 3 (35 points) Central Limit Theorem 3 0 1) 10 pts) A fair die is tossed.

Question

Problem 3 (35 points) Central Limit Theorem 3 0 1) 10 pts) A fair die is tossed. Let X be the number of dots on the side facing up. The sample space Ss(1,2,3,4,5,6), the probability mass function px(k)=1/6, k=1,2,3,4,5,6. Find the mean of X, and variance VAR[X]. 2) (10 pts) If a fair die is tossed 20 times, then the total of dots Y-X+X2tX20 where X i =1,2.20) are independent identical distributed (iid) random variables with mean E(X] and variance VARIX] (calculate in(1)). Using Central Limit Theorem, Y can be estimated as a Gaussian random variable. Find the approximate Gaussian distribution pdf fyly) for Y (15 pts) Using the result of 2), calculate the probability that the total number of dots is between 60 and 80 if a fair die is tossed 20 times. 3)

Explanation / Answer

Question 1:

As each of the face ont he dice here is equally likely, therefore the expected value for X here is computed as:

E(X) = (1/6)*(1 + 2 + 3 + 4 + 5 + 6) = 3.5

Therefore E(X) = 3.5

The second moment of X here is computed as:

E(X2) = (1/6)*(12 + 22 + 32 + 42 + 52 + 62) = 15.1667

Therefore, Var(X) = E(X2) - [ E(X)]2 = 15.1667 - 3.52 = 2.9167

Therefore V(X) = 2.9167

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.