The full quiz is here. The answers appear below. Comments, which are not part of the answers, are italicized.
1. Plot the PDF of an N(-10, 10) distribution. Clearly label the X and Y axes.
The shape of the distribution is easy: it's a "bell curve.". The plotting issues concern (a) where to center the curve on the X-axis, (b) how spread out along the X-axis it should be, and (c) how high it must be. The center (a) is at -10, the spread (b) is 10 times the spread of the standard normal distribution N(0, 1), which is essentially 0 when |X| > 3, and (c) to keep the area equal to 1.0, the height must be 1/10 the height of the standard normal distribution, because N(-10, 10) is ten times more spread out in the horizontal direction. The standard height (from tables or from the calculation of 1/sqrt(2*Pi)) is about 0.4 = 40%. Therefore the maximum height of N(-10, 10) is very close to 4.0%.
2. A distribution is an equal mixture of a Binomial(1, 0.8) distribution and a uniform distribution supported on [-1, 1]. Plot its CDF.
Let's begin by plotting the individual CDFs:
| Binomial (1, 0.8) CDF | Uniform(-1, 1) CDF |
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The mixture is the average of these two. It must contain a vertical leap of 0.2/2 = 0.1 at 0 and another leap of 0.8/2 = 0.4 at 1. Otherwise, it must increase linearly between -1 and 1, with a total contribution from the uniform component of 50%. Half of that rise, or 0.25, is achieved by the time X=0. Therefore the CDF equals 0 at X=-1, 0.25 at X=0, rises to 0.35 immediately, increases to 0.60 at X=1, then leaps to X=1.
3. A scientist assumes the weight of an adult male is Normally distributed with mean 70 Kg and standard deviation 10 Kg. What percentage of the adult male population weighs more than 55 Kg?
55 Kg = (55 - 70)/10 = -1.5 sd below the mean. The percentage therefore equals 1 - the value of the standard Normal CDF at Z = -1.5. We can estimate this value by remembering that 68% of the population will be between -1 and 1 sd and 95% between -2 and 2 sd. Therefore, because the Normal distribution is symmetric about its mean, about 34% is between -1 and 0, while about 48% is between -2 and -1 sd. Interpolating linearly gives (34 + 48)/2 = 41% is between -1.5 and 0 sd. Obviously 50% is greater than 0 sd. Therefore the desired value is about 50 + 41% = 91%. A more accurate answer is 93.32%, obtained from tables or a calculator.
4. Compute ln(1.000001) to sixteen decimal places. Show your work.
ln(1.00001) = ln(1 + 0.000001) = 10-6 - 10-12/2 + 10-18/3 - ... = 10-6 - 5*10-13 (to 18 decimal places) = 0.0000009999995000 (to 16 decimal places).
Scoring: The passing score is 95.
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This page was created 18 February 2001 and last modified 26 February to make the statement of problem 2 consistent with the statement in the quiz itself.