11-18. (Computer Problem for Normal Distribution)
There are numerous web sites which use numerical methods to calculate the area
Φ(x) =∫x
−∞
√^1
2 πe−x^2 / (^2) dx under the normalized probability curve φ(x) = √ 1
2 πe
−x^2 / (^2). Use
one of these web sites to verify the values given in the table 11.4.
11-19. (Binomial Distribution)
Show the variance of the binomial distribution f(x), given in the problem 11-11,
is σ^2 =npq by verifying the following relations.
(i) Show σ^2 =E[(x−μ)^2 f(x)] = E[x^2 ]−(E[x])^2
(ii) Show E[x^2 ] =
∑nx=1[x(x−1) + x]f(x) = n(n−1)p^2 +np(iii) Show σ^2 =npq
11-20. Sketch the bell shaped probability density curve φ(z) = √^12 πe−z^2 /^2 associ-
ated with the normalized normal distribution. Find and show sketches of the given
probabilities as areas of shaded regions on this curve.
(a)P(z≤ 0 .5 )
(b)P(z≤− 2 .2 )
(c)P(|z|≤ 2 )(d)P(z > 0 .5 )
(e)P(− 1. 3 ≤x≤ 0 .76 )
(f)P(|z|≤ 3)(g)P(z < 2 .3 )
(h)P(z > 2 .3)
(i)P(|z|≤ 1 )11-21. (Hypergeometric distribution)
A certain shipment of transistors contains 12 transistors, 3 of which are defective.
If a batch of 5 transistors is drawn from the shipment, then
(a) determine f(x), x = 0 , 1 , 2 , 3 which represents the probability that xof the 5 items
selected are defective.
(b) determine the minimum number of transistors that must be drawn to make the
probability of obtaining at least 5 nondefective transistors greater than 0.8.
11-22. Let Xdenote a random variable normally distributed with mean μ= 12 and
standard deviation σ= 4. Find values of and show with sketches the probabilities as
areas of shaded regions for representing the probabilities
(a) P(X≤14)
(b) P(8 ≤X≤16)(c) P(X≤10)
(d) P(0 ≤X≤24)