Question

2. Honest data is repeatedly released independently. Let Xi be the result of i-th launch and Sn = X1 + X2,. . . , Xn, obtain:

(Letter b needs to be equal to 0,9608. Please, show me the reasoning)

a) lim→∞ P(Sn> 3n).
b) An approximate value for P (S100> 320) 

Answer 1

A)

lim→∞ P(Sn> 3n).

lim→∞ (3n)=3∞ =1

B)Let us call N320 this number. We want to calculate P(N320 < 100). There is
no obvious way of expressing the random variable N320 as the sum of independent
random variables, but we can proceed differently. Let Xi be the processing time
of the ith part, and let S100 = X1 + ··· + X100 be the total processing time of the
first 100 parts. The event {N320 <100} is the same as the event {S100 >320},
and we can now use a normal approximation to the distribution of S100. Note that
µ = E[Xi] = 3 and σ2 = var(Xi) = 16/12 = 4/3. We calculate the normalized
value
z = 320 − nµ/σ√n = 320 − 300/√(100 .4/3)
= 1.73,
and use the approximation
P(S100 > 320) ≈ Φ(1.73)=0.9582.