The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 3 < X < 5 ) = \frac { 1 } { 8 }\) and \(\mathrm { E } ( X ) = 4\)
find the value of \(a\) and the value of \(b\)
find the value of the constant, \(c\), such that \(\mathrm { E } ( c X - 2 ) = 0\)
find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\)
find \(\mathrm { P } ( 2 X - b > a )\)
Left-handed people make up \(10 \%\) of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
Write down an expression for the exact value of \(\mathrm { P } ( Y \leqslant 1 )\)
Evaluate your expression, giving your answer to 3 significant figures.
Using a Poisson approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
Using a normal approximation, estimate \(\mathrm { P } ( Y \leqslant 1 )\)
Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm { P } ( Y \leqslant 1 )\)