Let X1 X2 Y10 be independent identically distributed iid co

Let X_1, X_2, ..., -Y_10 be independent identically distributed (i.i.d) continuous random variables with E[X_i] = 1 and Vnr[X_2]=l for i = 1,2, ..., 10. Find the expected value E[M10] and the variance of the sample mean Provide an upper-bound on the probability that the random variable M_10 exceeds 4 or is below -2. Use the central limit theorem approximation to obtain an \"approximation\" of the probability that the random variable M_10 deviates from its mean by more than.

Solution

M₁₀ is the mean of the 10 identically distributed variables.

E(Xi) = 1 and var(xi) = 1

E(M₁₀) = 1

and var(M₁₀) = Var(xi)/n = 0.1

Hence M₁₀ is normal with (1, 0.316)

2) P(M₁₀>4 or M₁₀<-2) = P(Z>9.486 or z<-9.486)

=0

3) P(More than 3 sigma) = P(|Z|>3) = 0.0026