Scientists concerned with environmental issues are interested in the sources of acid rain. Nitrates are a constituent of acid rain with arsenic an accompanying element.

It can be assumed that the variables X (nitrate concentration in micromoles per litre) and Y (arsenic concentration in nanomoles per litre) follow the following bivariate normal distribution

(X) ~ N2 [(49.8) , ($\displaystyle 30.7^2$ 26.52)]

(Y) ~ N2 [(2.26) , (26.52 $\displaystyle 1.18^2$)]

I hope the way the bivariate distribution is set out is comprehendable, i was having severe problems with the LaTex for it. Hopefully people can picture how a bivariate distribution is usually set out and can translate this accordingly!

(i) What are the marginal distributions of X and Y?

(ii) Identify Corr(X,Y) and explain what it means in terms of the make up of acid rain.

(iii) Obtain the distribution of Y|X = x and use it to determine E[Y|X = x] for a nitrate concentration of 60 micromoles per litre.