Question

1.a)Pregnancy lengths are normally distributed with a mean of 39 weeks and a standard deviation of 2 weeks. Find the following.

a. P(X > 36)

b. P(37 < X < 40)

c. The cutoff for the shortest 10% of all pregnancy lengths

1.b)Suppose the random variable X has a Poisson distribution with rate L use the moment generating function method to show that the mean and variance of X are both L.

1.c)Let X have a gamma distribution. a. Use the integral definition to find E(X^ 2 ). b. Use the moment generating function method to find E(X ^2 ).  

Answer 1

Solution :

(a)

P(x > 36) = 1 - P(x < 36)

= 1 - P[(x - ) / < (36 - 39) / 2]

= 1 - P(z < -1.5)

=0.0668

.

(b)P(37 < x < 40) = P[(37 - 39)/ 2) < (x - ) /  < (40 - 39) / 2) ]

= P(-1 < z < 0.5)

= P(z < 0.5) - P(z < -1)

= 0.6915 - 0.1587

= 0.5328

(c)

P(Z < -1.28) = 0.10

Using z-score formula,

x = z * +

x = -1.28 * 2 + 39 = 36.44

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