Customers arrive at a service facility according to a Poisson process of rate λ customers/hour. Let X(t) be the number of customers that have arrived up to time t. Let W1,W2,... be the successive arrival times of the customers. (a) Determine the conditional mean E[W1|X(t)=2].
(b) Determine the conditional mean E[W3|X(t)=5].
(c) Determine the conditional probability density function for W2, given that X(t)=5.

Farmers know that driving heavy equipment on wet soil compresses the soil and injures future crops. here are data on the "penetrability" of the same type of soil at two levels of compression. penetrability is a measure of how much resistance plant roots will meet when they try to grow through the soil. compressed soil 2.85 2.66 3 2.82 2.76 2.81 2.78 3.08 2.94 2.86 3.08 2.82 2.78 2.98 3.00 2.78 2.96 2.90 3.18 3.16 intermediate soil 3.17 3.37 3.1 3.40 3.38 3.14 3.18 3.26 2.96 3.02 3.54 3.36 3.18 3.12 3.86 2.92 3.46 3.44 3.62 4.26 use the data, omitting the high outlier, to give a 95% confidence interval for the decrease in penetrability of compressed soil relative to intermediate soil. compute degrees of freedom using the conservative method. interval: to