Quiz #3

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Question 1 B
Question 2 D

Question 1

If X is a normally distributed random variable having mean 50 and variance 25, what is the probability that it will assume a value less than 45? (Choose the best answer from the choices given below. Due to rounding, none of them is exactly equal to the desired probability.) The probability that a normal random variable assumes a value within 1 standard deviation of its mean is about 0.68, and so the probability that a normal random variable assumes a value one or more standard deviations away from the mean is about 0.32. Since the density of a normal distribution is symmetric about its mean, the probability that a normal random variable assumes a value more than 1 standard deviation below its mean is about 0.16, and the probability that a normal random variable assumes a value more than 1 standard deviation above its mean is about 0.16. Since the standard deviation of X is 5, less than 45 is less than one standard deviation below the mean of X, and so the desired probability is (about) 0.16.

Question 2

Suppose a random variable V has the density which is sketched on the board (once I get it up there). What is the value of P(V > 0.5)? V has possible values between 0 and 1. If the density of V was constant (i.e., if V was uniformly distibuted), then the desired probability would be 0.5. But the density of V is greater for values above 0.5 than it is for values below 0.5, and so the desired probability is greater than 0.5. Clearly the probability is not 1 (or 2), and so the only reasonable answer is 0.75. Geometry can be used to show that the probability is exactly 0.75 --- the area under the density over (0, 0.5] is clearly 0.25 (making use of the fact that the area of the triangle is one half the base times the height), and so the area under the density over (0.5, 1.0) must be 0.75.