6.8 #2
The area under this triangle is 1/2 base*height = 1/2*10*a = 5a. But it's supposed to be 1 (total probability = 100%), so a=1/5.
#3
Likewise, this area is 1/2*100*a = 50a, so a=1/50.
#6
The area from x=0..50 is 50*2c. The area from x=50..75 is 25*c. The total is 125c. To make this 1, c=1/125.
#10
The slope of this line is -0.1/20. So the function p(x) = 0.1 - 0.1/20*x. You could do the integrals from there, or just argue geometrically; the average of a linear function over some interval is the average of the values at the endpoints.
#12Any function that is only nonzero between x=0 & 200, and is bigger on the right side than the left (with total area 1).
#13
Reverse the above, but only nonzero between x=0 & 30.
#16
People are much more likely to stop their education at or near certain times, namely 8, 12, or 16 years. (Actually this graph is a bit ridiculous because it's so smooth, as if there were a certain number of people stopping at 11.693 years of education. In fact they just took a list of numbers (one for each year) and fit a smooth curve to them.)
Estimate the percentage with less than 10 years: this is the ratio of the area from 0 to 10 to the total area. Both the 8 and the 12 peaks are roughly triangular, and the 12 peak is about 2.5 times as high as the 8 but the same width. Plus there's the 16 peak, so let's lump the 16 in with the 12 and decide that they make a peak as wide as the 8 peak but 3 times as high, so three times as much area. Therefore the first peak only has about 25% of the area.
#17
This is a little hard to answer without deciding what "most people" and "near 50" means. One could ask instead: for what number X did half of the people get a score between 50-X and 50+X?
If we take X=2, we're calculating the area of that peak, basically - very small, not nearly half the people. Even X=20 is only getting the area between 30 and 70, which looks like a bunch less than the area between 0 and 30. So yeah, 50 was a popular number, but not nearly popular enough to say that "most people" (which one would generally interpret as >50%) got "near" it (presumably >20 points off is not really "near").