Probstat/week8 practice 2

This is part of probstat.

To compute the cdf of standard normal random variable, you can use a table from wikipedia.

Basic practice

Let Z be a standard normal random variable (whose mean is 0 and variance is 1).

1. Find P{ Z >= 0 }.

2. Find P{ Z >= 0.5 }.

3. Find P{ Z >= 1 }.

4. Find P{ Z >= -0.3 }.

5. Find P{ -1 <= Z <= 0.1 }.

6. Find P{ Z <= 0.2 }

7. Approximate the minimum value a such that P{ Z >= a } <= 0.1.

8. Approximate the minimum value a such that P{ -a <= Z <= a } >= 0.7.

9. The download speed of a file to your computer is normally distributed with parameters $\mu =5$ Mbps and $\sigma ^{2}=0.5$. What is the probability that you get the download speed faster than 6 Mbps? What is the probability that your speed is below 4 Mbps?

10. The average height that Somchai the athlete can jump is 2 m with variance 0.16. Today to win, Somchai has to jump higher than 2.4 m. What is the approximate probability that Somchai will win?

Challenging problems

1. There are 2 types of fishes in the pond: type A and type B. The average length of type-A fish is 50 cm with variance 10. For type B, the average length is 65 cm with the same variance. You want to classify these two types of fishes using just their lengths.

• 1.1 Plot the density functions of the lengths of fish of type A and of type B. Plot them in the same graph (using different types of lines).
• 1.2 You want to classify the fishes using a single threshold L = 60. That is, if you pick a random fish, and its length is at most L, you will say that it is type A, and you will say that it is type B, otherwise. What is the probability that the fish that you pick is actually of type A, but you classify it wrongly as type B.
• 1.3 Using the threshold L = 60, what is the probability that the fish that you pick is actually of type B, but you classify it wrongly as type A.
• 1.4 From the given information, can you find the probability that you make a wrong classification that using L=60 as the threshold? If you can't, what information is missing?
• 1.5 The threshold L=60 might not be the best. What is the best threshold value L that you can use to classify the fishes?
• 1.6 Using the threshold that you proposed, what is the probability that the fish that you pick is actually of type A, but you classify it wrongly as type B.
• 1.7 Using the threshold that you proposed, what is the probability that the fish that you pick is actually of type B, but you classify it wrongly as type A.
• 1.8 From the given information, can you find the probability that you make a wrong classification that using the threshold that you proposed? What is the difference between this situation and the situation in question 1.4.