Question

1) A paper reports the number of defective transistors across a square centimeter of an integrated circuit follows the Poisson distribution with λ = 100. But you're interested in the distribution of defective transistors over half that area. What value should λ be?

Answer is 50 but I don't understand why.

2) Wine is considered properly handled if the stored temperature is within 1 standard deviation of the mean. If wine is shipped in batches of 20 bottles, in the long run what proportion of shipments will have at least 19 bottles properly handled?

Answer:

pr(x ≥ 19) = p(19) + p(20) = 20π^19 -20π^20 + π^20 (you get this using the binomial distribution with n=20)

to find π, my professor said π = the proportion of successfully handled bottles (which I understand) = area under Normal within 1 st. deviation of the mean = 0.68.

I do not understand the bolded part of that statement or the logic used to make those steps in order to get that π = 0.68.

Answer 1

A paper reports the number of defective transistors across a square centimeter of an integrated circuit follows the Poisson distribution with λ = 100.

Next we want to find the rate of defective transistors over half area.

So the new parameter is = old parameter*(area proportion) = 100*(1/2) = 50

2) Wine is considered properly handled if the stored temperature is within 1 standard deviation of the mean.

Let X be the stored temperature to handled the Wine.

So the probability of a bottle which are properly handle = P( X lies between one standard deviation from mean)

= P( -1 < Z < 1) = P(Z < 1) - P(Z < -1) = "=NORMSDIST(1)-NORMSDIST(-1)" = 0.6827 = 0.68

(note that here we use excel command.

Which is constant for each trial.

n = 20 = total number of bottle in inspection.

Here each trial is independent to the others.

Let X = number of bottles handled properly.

So we can use the binomial distribution to find the probability of how many bottles handled properly.

If wine is shipped in batches of 20 bottles, in the long run what proportion of shipments will have at least 19 bottles properly handled?

That is here we want to find P(X >= 19 ) = 1 - P(X <= 18)

Let's use excel:

P(X >= 19 ) = "=1-BINOMDIST(18,20,0.68,1)" = 0.0047