Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 4, Section 3



  1. In class we drew the probability histogram for the random variable X = the sum of pips on two dice. Suppose you rolled two dice a thousand times, kept careful record of the outcome of each roll, and plotted the relative frequencies of a `2', `3', ..., `12' in a histogram. How would you expect your histogram to compare to the probability histogram from class?

    The probability histogram as a mathematical model, arising from the assumptions that the roll you get on one occasion is independent of previous rolls and that all rolls are equally-likely. For dice that satisfy these assumptions, one would expect that a relative-frequency histogram of actual rolls would tend to settle in to the appearance of the probability histogram as the number of rolls n increased. After n=1000 rolls, the “actual” histogram is libel to look very much like that of the probability model.

  2. Suppose the random variable X = SAT-math score. Explain why X is a discrete random variable.

    It is discrete because scores are always integers. There might be people with scores of 550 and others with 551, but no one gets a score in between these two (or any other pair of) values.

  3. Normal distributions are continuous probability distributions. If SAT-math scores have a discrete probability distribution (X in the previous question is, after all, a discrete random variable), then was I in error when I made the statement in class that SAT-math scores are distributed normally as N(500,100)?

    Literally speaking, I was in error. It is the same kind of error we make when we say that the number p is equal to 22/7. The discrete probability distribution for SAT-math scores is very similar in shape to N(500,100) - the latter is roughly what you would see if you blurred your eyes when looking at the former, blurring them enough so as not to see the stair-step effect of the bars at each possible SAT-math score. Moreover, since probabilities are illustrated as areas under these probability-distribution curves, the most important thing is that, for a give value of x, both graphs have approximately the same amount of areas to give values for P(X < x).