The random variable \(X\) has mean 17 and variance 64 . The independent random variable \(Y\) has mean 10 and variance 16 . Find the value of
\(\mathrm { E } ( 4 Y - 2 X + 1 )\),
\(\quad \operatorname { Var } ( 4 Y - 5 X + 3 )\),
\(\mathrm { E } \left( X ^ { 2 } Y \right)\).
For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \(( 19,20 )\).
Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4 x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\).
State two reasons why your prediction in part (a) may not be reliable.
It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
Find the probability that the lifetime of a randomly selected hair dryer is between 1.8 and \(2 \cdot 5\) years.
Given that \(20 \%\) of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\).
Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers.
State one assumption that you have made in part (c).
A continuous random variable \(X\) has cumulative distribution function \(F\) given by
$$F ( x ) = \begin{cases} 0 & \text { for } x < 0 \frac { 1 } { 4 } x & \text { for } 0 \leqslant x \leqslant 2 \frac { 1 } { 480 } x ^ { 4 } + \frac { 7 } { 15 } & \text { for } 2 < x \leqslant b 1 & \text { for } x > b \end{cases}$$
Show that \(b = 4\).
Find \(\mathrm { P } ( X \leqslant 2 \cdot 5 )\).
Write down the value of the lower quartile of \(X\).
Find the value of the upper quartile of \(X\).
Find, correct to three significant figures, the value of \(k\) that satisfies the equation \(\mathrm { P } ( X > 3 \cdot 5 ) = \mathrm { P } ( X < k )\).