Question

5. The maximum patent life for a new drug is 17 years. Subtracting the length of time

required by the FDA for testing and approval of the drug provides the actual patent life

of the drug – that is, the length of time that a company has to recover research and

development costs and make profit. Suppose the distribution of the lengths of patent life

for new drugs is as shown here:

x 3 4 5 6 7 8 9 10 11 12 13

f(x) 0.03 0.05 0.07 0.10 0.14 0.20 0.18 0.12 0.07 0.03 0.01

a) Find the expected number of years of patent life for a new drug.

b) Find the standard deviation of x.

c) Find the probability that x falls into the interval μ ± 2σ.

Answer 1

$a:\\EX=\sum{xf\left( x \right) =}7.9\\b:\\EX^2=\sum{x^2f\left( x \right)}=67.14\\DX=EX^2-\left( EX \right) ^2=67.14-7.9^2=4.73\\\sigma X=\sqrt{DX}=\sqrt{4.73}=2.17486\\c:\\P\left( 7.9-2\cdot 2.17486\leqslant X\leqslant 7.9+2\cdot 2.17486 \right) =\\=P\left( 3.55028\leqslant X\leqslant 12.2497 \right) =P\left( 4\leqslant X\leqslant 12 \right) =\\=1-P\left( X=3 \right) -P\left( X=13 \right) =1-0.03-0.01=0.96$