QSCI381: Introduction to Probability & Statistics
Problem set #4
#1. Determine whether the following can be probability distributions. Explain why?:
a) P(X) = 1/5 for X = 0, 1, 2, 3, 4, 5.
b) P(X) = (X-2)/5 for X = 1, 2, 3, 4, 5.
c) f(X) = 4X for 0 £ X £ 1.
#2. Suppose the diastolic blood pressure (X) in hypertensive women centers about a mean of 100-mm Hg with a standard deviation of 14-mm Hg and is normally distributed. Find:
a) P(96 < X £ 110)
b) P(X < 88)
c) P(X > 97)
d) xo such that P(X £ xo) = 2/3
e) xo such that P(X > xo) = 1/5
f) The value k such that P({100-k} < X < {100+k}) = 0.90
#3. Assuming that 10% of fish caught in Puget Sound suffer from red-tide disease, what is the probability that at least one of the 15 fish caught will be diseased.?
#4. Find the probability of getting the first hit on the rth try, when the probability of scoring a hit is 0.8 and remains constant on each try. Do this for r £ 6. Compute the mean and variance for this probability distribution and compare it with the mean (m = 1.25) and variance (s 2 = 0.3125) for the GEOMETRIC distribution with P = 0.8
#5. Following data was collected during a survey of nursery beds to detect the incidence of plant infection:
|
Number of infected plants per bed (Xi) |
0 |
1 |
2 |
3 |
4 |
5 |
|
Number of beds (Ni) |
600 |
310 |
80 |
7 |
2 |
1 |
a) Compute the mean and variance for this distribution and check if they are approximately equal, as is the case with Poisson probability distribution.
b) Calculate expected frequencies if the infection actually followed Poisson distribution with mean (m ) = 0.5.
#6. The heights of students on this campus are normally distributed. If 10% of the students are at least 72.5 inches tall and 5% of the students are no taller than 62.2 inches, what is the mean and standard deviation of this distribution?
#7. Using normal approximation, compute the probability of observing the following number of heads (equivalent to observing 0.49 £ p £ 0.51) for given number of tosses:
a) between 49 and 51 when a balanced coin is tossed 100 times;
b) between 490 and 510 when a balanced coin is tossed 1000 times;
c) between 4900 and 5100 when a balanced coin is tossed 10,000 times
Why do you think there are differences among these probabilities?.