## Mathematics 210 - Spring term 2003 - fifth assignment

This assignment requires you to submit several spreadsheets concerned with probability distributions, particularly approximation by the normal distribution. It is dfue on Monday, February 24. I repeat the instructions from the first lab:
• Question 1. Make a table of values of the standard normal distribution y = (1/) e-x2/2 in the range 0 to 6, with an interval of 0.1. Then make a table of the integral of the function in the same range, using Simpson's rule. Graph both functions on the same graph.

Save this sheet as m210.5.1.ms.

• Question 2. Use that table to estimate the area A(0,x) under the normal curve between 0 and x for x = 0.256, x=1.376, x=1.678. Use linear interpolation

f(a + sh) = (1-s)f(a) + sf(b)

to do this, where h=(b-a). Put the answers on the graph, clearly indicated and labeled. Then use linear interpolation to estimate the value of x for which A(0,x) is 1/4.

Save this sheet as m210.5.2.ms.

• Question 3. Answer the same question, but this time using cubic interpolation. Here you will have to realize that dx/dy = 1/(dy/dx). Remembr that if values of x are in one column and f(x) in another, then the derivative is roughly the ratio of changes in the second to the changes in the first. Going backwards inverts this.

f(a + sh) = (1-s)3f(a) + 3(1-s)2s[f(a)+(1/3)f'(a)h] + 3(1-s)s2[f(b)-(1/3)f'(b)h] + s3f(b)

where h = b-a.

Save this sheet as m210.5.3.ms.

• Question 4. Plot on the same graph as a plot of the calculated A(0, x) the first three asymptotic formulas taken from
A(x, infinity) = (1/) e-x2/2 (1/x - 1/x3 + 3/x5 - 3.5/x7 + ... )

for A(0, x), for x in the range 1 to 6. label them clearly. (You might want to take a look at my notes on this formula.)

Save this sheet as m210.5.4.ms.

• Question 5. Suppose that 1,000 numbers are chosen randomly and independently from an interval of [0, 1]. Graph the bell curve that approximates the distribution of their sum. Graph it, scaling it sensibly. (If m is the mean of the distribution of a single number and v its variance, the probability distribution for the sum is very closely approximated by the normal distribution with mean 1000 m and variance 1000 v.) Estimate the range, centered at the mean, within which half the sums fall.

Save this sheet as m210.5.5.ms.

That's it!

If you find these questions confusing, please write me.