| a,i) III | a,ii) VI |
| b,i) I | b,ii) V |
| c,i) IV | c,ii) II |
#4
Density: something that peaks in the middle (class). Cumulative: something logistic-y, increasing slowly, then quickly, then slowly, up to 1.
#5
Density: start large and drop to a small value, then stay low. Cumulative: increase a lot first, then increase little, up to 1.
#6
Density: start large, go low, go high again (then go to zero). Cumulative: increase a lot, then stay nearly flat, then increase a lot, suddenly stopping at 1.
#8
The integral from A to B is the probability that somebody lives exactly that long (i.e. between A and B). Put differently, it's the probability that they die at a time in between A and B. So if A is zero, it's the probability that they die before time B. So P(t) = integral from 0 to t = the fraction that have died by time t. Note that it goes to 1 - everybody eventually dies.
#16
a) The function is 0, then some constant for x=0..5, then 0 again. We almost know it; all that's left is to determine the constant. All else we know is that the area is 1 - so the constant is 1/5. Therefore f(r)=1/5 for r=0..5, 0 otherwise.
b) Integrating that, we get a function that's 0 for r<0, r/5 for r=0..5, and 1 for r>5.
6.10#3
a) Where does this cross y=1/2? Something like x=32.
b) Call the cumulative function C. The density, C', is positive between 0 and 60 (C is always increasing, until it makes it up to 1). It's increasing where C is concave up, between 0 and 36 or so. It's decreasing where C is concave down, between 36 or so and 60. C gets most concave up at about 32, and most concave down at 45; these are the maxima and minima of C'.
#6
a) Typical exponential decrease.
b) The indefinite integral is just -e^(-0.1t)+C, so the definite integral from 0 to x is 1-e^(-0.1x). For this to be 1/2 (where we get the median), we have