- The random variable \(X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)\) and the random variable \(Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)\)
\(X\) and \(Y\) are independent random variables.
A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)
The random variable \(W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y\) and has the distribution \(\mathrm { N } \left( 169,2 ^ { 2 } \right)\)
Find the value of \(a\) and the value of \(b\)