A large but sparsely populated county has two small hospitals, one at the south end of the county and the other at the north end
. The south hospital’s emergency room has four beds, whereas the north hospital’s emergency room has only three beds. Let X denote the number of south beds occupied at a particular time on a given day, and let Y denote the number of north beds occupied at the same time on the same day. Suppose that these two rv’s are independent; that the pmf of X puts probability masses .1, .2, .3, .2, and .2 on the x values 0, 1, 2, 3, and 4, respectively; and that the pmf of Y distributes probabilities .1, .3, .4, and .2 on the y values 0, 1, 2, and 3, respectively.a. Display the joint pmf of X and Y in a joint probability table.b. Compute P(X # 1 and Y # 1) by adding probabilities from the joint pmf, and verify that this equals the product of P(X # 1) and P(Y # 1).c. Express the event that the total number of beds occupied at the two hospitals combined is at most 1 in terms of X and Y, and then calculate this probability.d. What is the probability that at least one of the two hospitals has no beds occupied?Devore, Jay L.. Probability and Statistics for Engineering and the Sciences (Page 210). Brooks Cole. Kindle Edition.
1 answer:
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Answer:
1. The probability that x will be more than 25 is 0.6305.
2. The 55th percentile is 26.38.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean 26 and standard deviation 3.
This means that
1 Find the probability that x will be more than 25.
This is 1 subtracted by the p-value of Z when X = 25. So
has a p-value of 0.3695.
1 - 0.3695 = 0.6305.
The probability that x will be more than 25 is 0.6305.
2 Find the 55th percentile of x.
This is X when Z has a p-value of 0.55, so X when Z = 0.125.
The 55th percentile is 26.38.
As we have to solve for b, it means we have to isolate b
So first of all subtract from both sides
So we get
Now we need b , but its b^2 , so we can take square root on both sides.
taking square roots on both sides
Hence option B is correct