A couple of Commone Probability Distribution Problems

  1. A countinuous uniform distribution has the parameters a=4 and b=10. The F(20) is: A. 0.25 B. 0.50 C. 1.00 D. 2.00 2. About 50% of all observations for a normally distributed random variable fall in the interval: A. mean ± 0.67ó B. mean ± ó C. mean ± 2ó D. mean ± 3ó How’s your thinking process to deal with this type of questions? Thx.

For the first Q, the answer is C. The cumulative distribution function F(x) is 0 if x=b, where a and b are the limits. In this case, x=20 is outside the interval, so the only answer is C, F(20)=1. For the second Q, remember that 68% of observations are in 1stdv around the mean, 95% in about 2stdv, 99% in about 3stdv. You’re looking for 50% f the observations, which would be less than 68%, which is less than 1stdv. Intuitively it should be A. To make sure, check the tables.

My thoughts would be as follows: 1) F(20) - since 20 > 10 (the upper bound), it must be equal to 1. 2) Well, 1 standard deviation is approximately equal to 67%. Hence, it must be less than one standard deviation. Therefore, the answer should be A. Hope im correct?

Thanks, now I see how obviously it is… For the 1st Q, I was stuck at the format of the question. I think it’s actually asking what’s the probability of a random variable greater than 20? P(X>20) = 1 - P(X<=20) = 1 - F(20) = 0 (because P(X) shall fall in the range of a F(20) = 1 For the 2nd Q, I originally thought there might be a way to calculate the exact confidence interval. From your reply, it seems it’s just an implied guess from what we already know.

Not really a guess if you check the tables. You need the z value that leaves 25% of the observations in the right tail, and 75% o the left. Search for 0.75 in the table for cumulative probability for a standard normal distribution. That happens for a z between 0.67 (for which 0.7486 are in the left tail) and 0.68 (0.7517). The closest z is in answer A (0.67) compared to answers B (z=1), C(z=2), and D (z=3).

Thx. I’m wondering why not the symetric percentage from the mean to both sides?

it’s not a guess at all. we know that 68% of observations fall within +or- 1 SD. therefore 50% of observations must be less then 1 SD.