2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)
- Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
- Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\)
The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
- Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\)
The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
- Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)