91Ó°ÊÓ

Temperature is the degree or intensity of heat present in a substance or system, as measured by a thermometer and expressed on a comparative scale.

If we use the random variable\({\rm{X}}\)to represent the temperature in degrees Celsius, we can deduce that

\(\begin{array}{l}{{\rm{\mu }}_{\rm{X}}}{\rm{ = 120}}\\{{\rm{\sigma }}_{\rm{X}}}{\rm{ = 2}}\end{array}\)

It is also stated that the temperature in degrees Fahrenheit (\(^{\rm{^\circ }}{\rm{F}}\)) and degrees Celsius (\(^{\rm{^\circ }}{\rm{C}}\)) are connected.

\(^{\rm{^\circ }}{\rm{F = 1}}{\rm{.}}{{\rm{8}}^{\rm{^\circ }}}{\rm{C + 32}}\)

Now, if we use the random variable\({\rm{Y}}\)to represent the temperature in degrees Fahrenheit, we may write,

\(\begin{aligned}{{\rm{\mu }}_{\rm{Y}}} &= 1{\rm{.8(120) + 32}}\\ &= 248 \\{{\rm{\sigma }}_{\rm{Y}}} &= \sqrt {{{{\rm{(1}}{\rm{.8)}}}^{\rm{2}}}{\rm{ \times (2}}{{\rm{)}}^{\rm{2}}}} \\ &= 3 {\rm{.6}}\end{aligned}\)

The expected value and variance of\({\rm{h(X)}}\)satisfy the following conditions for\({\rm{h(X) = aX + b}}\):

\(\begin{aligned}E(h(X)) &= a\mu + b\\V(h(X)) &= {{\rm{a}}^{\rm{2}}}{{\rm{\sigma }}^{\rm{2}}}\end{aligned}\)

Therefore, the values are \({\rm{248}}\) and \({\rm{3}}{\rm{.6}}\).