S2 January 2013 Q5

EdexcelOld spec10 marksContinuous Random Variables

5. The continuous random variable \(T\) is used to model the number of days, \(t\), a mosquito survives after hatching.

The probability that the mosquito survives for more than \(t\) days is

\[\frac{225}{(t + 15)^2}, \qquad t \geqslant 0\]

(a) Show that the cumulative distribution function of \(T\) is given by \[\mathrm{F}(t) = \begin{cases} 1 - \dfrac{225}{(t + 15)^2} & t \geqslant 0 \\ 0 & \text{otherwise} \end{cases}\] (1)

(b) Find the probability that a randomly selected mosquito will die within 3 days of hatching. (2)

A large number of mosquitoes hatch on the same day.

(d) Find the number of days after which only 10% of these mosquitoes are expected to survive. (4)

Mark scheme (a)

Scheme	Marks
\(\mathrm{P}(T \gt t) = \dfrac{225}{(t + 15)^2}\)
\(\mathrm{P}(T \leqslant t) = 1 - \mathrm{P}(T \gt t)\) \(= 1 - \dfrac{225}{(t + 15)^2}\)
\(\mathrm{F}(t) = \begin{cases} 1 - \dfrac{225}{(t + 15)^2} & t \geqslant 0 \\ 0 & \text{otherwise.} \end{cases}\)	B1
	(1)

Notes

B1 The line \(\mathrm{P}(T \leqslant t) = 1 - \mathrm{P}(T \gt t)\) or \(\mathrm{F}(t) = 1 - \mathrm{P}(T \gt t)\) or both of the following statements \(\mathrm{P}(T \gt t) = \dfrac{225}{(t + 15)^2}\) and \(\mathrm{P}(T \leqslant t)\) /\(\mathrm{F}(t) = 1 - \dfrac{225}{(t + 15)^2}\) must be seen and no errors. Allow equivalent in words.

Condone use of \(\lt\) instead of \(\leqslant\) or \(\gt\) instead of \(\geqslant\) and vice versa.

The cdf must be given. Allow \(t \gt 0\)

Scheme	Marks
\(\mathrm{P}(T \lt 3) = 1 - \dfrac{225}{(3 + 15)^2}\)	M1
\(= \dfrac{11}{36}\) or \(0.30555\ldots\) awrt 0.306	A1
	(2)

Mark scheme (c)

Scheme	Marks
\(\mathrm{P}(T \gt 8 \mid T \gt 3) = \dfrac{P(T \gt 8)}{P(T \gt 3)}\)	M1 M1
\(= \dfrac{\;\frac{225}{23^2}\;}{\;\frac{225}{18^2}\;}\)
\(= \dfrac{324}{529}\) or \(0.612..\) awrt 0.612 / 0.6125	A1
	(3)

Notes

1st M1 The conditional probability must,

be a quotient and
have \(\mathrm{P}(T \gt 3)\) or ‘their numerical equivalent’ for the denominator and
have \(\mathrm{P}(T \gt 8)\) or \(\mathrm{P}(T \gt 5)\) or \(\mathrm{P}(T \gt 8 \cap T \gt 3)\) or \(\mathrm{P}(T \gt 5 \cap T \gt 3)\) or ‘their numerical equivalent’ for the numerator.

Allow \(\geqslant\) in place of \(\gt\)

2nd M1 writing or using \(\mathrm{P}(T \gt 8)\) or \(\mathrm{P}(T \geqslant 8)\).

NB This is independent of the first M mark.

Mark scheme (d)

Scheme	Marks
\(1 - \mathrm{F}(t) = 0.1\)	M1
\(\dfrac{225}{(t + 15)^2} = 0.1\) or \(1 - \dfrac{225}{(t + 15)^2} = 0.9\)	A1
\(\dfrac{225}{0.1} = (t + 15)^2\)
\(t = \sqrt{\dfrac{225}{0.1}} - 15\)	M1
\(t = 32.4\), also accept 32/33	A1
	(4)
	(10 marks)

Notes

1st M1 writing or using \(1 - \mathrm{F}(t) = 0.1\) or \(\mathrm{P}(T \geqslant t) = 0.1\) May be implied by \(\dfrac{225}{(t + 15)^2} = 0.1\) o.e.

2nd M1 either square rooting or solving a quadratic either by factorising / completing the square / using the formula - must be correct for their quadratic.

A1 awrt 32.4 or 32 or 33. Do not accept \(15\sqrt{10} - 15\)