State the sum rule of: i. The probability thet either event A or event B occurs.

Question

State the sum rule of: i. The probability thet either event A or event B occurs. ii. The probability for three independent events A, B and C. b) A company produces semiconductor laser devices. The threshold current I of its products, measured in mA, is normally distributed. It is found thet P(I 107) = 0.0735. Find the mean and standard deviation of the threshold current I. c) The laser devices with the threshold current I within the range 99 to 107 mA are acceptable while others are regarded as defective. A sample of 5 devices is chosen at random. Find the probability thet this sample contains i. exactly two defective devices, and ii. at least one defective device. Give your answers in 4 decimal places. d) The moment generating function G(t) is defined by G(t) = E(e^tx1) =sigma_i e^tif(x_i) or G(t) = E(e^tx) = integral^99_-99 e^x f(x)dx i. Prove thet F(X^k) = G(^k)(0), where G(^k) = d^kG/dt^t. In particular, show thet mu = G'(0). ii. Show thet the binomial distribution has the moment generation function G(t) = (pe^t +q)^n. iii. Show thet the Poisson distribution has the moment generation function G(t) = e^-ue^t iv. Using (ii). prove thet for a binomial distribution mu = np and sigma = npq. iv. Using (ii), prove thet for a binomial distribution mu and sigma^2 = npq. v. Using (iii), prove thet for a Poission distribution sigma^2 = mu

Explanation / Answer

a) Probability of either event A or event B occurs = P(A) +P(B) -P(A and B)

Probability of three independnet events of A, B and C = P(A and B and C) = P(A)*P(B)*P(C)

b) using the normal standard distribution

P(z<-1.04982) = 0.1469

since z=x- mean/sd therefore 99= mean + sd*-1.04982 -----(1)

from the second equation p(l<=107)=1-0.0735 =0.9265

also from standard normal tables P(z<=1.45021) = 0.9265

therefore 107= Mean + SD*1.54021 ------ (2)

Solving (1) and (2) to get the values of 'mean' and 'SD'

SD=3.2; Mean =102.36

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