13. Suppose that a certain examination is to be taken by five students independently of one another, and the num- ber of minutes required by any particular student to com- plete the examination has the exponential distribution for which the mean is 80. Suppose that the examination be- gins at 9:00 AM. Determine the probability that at least one of the students will complete the examination before 9:40 A.M.
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
question 13
![IS SUUI AS CIre u ue compsremS TaIs. Suppuse sU uKIT
the length of life of each component, measured in hours,
has the exponential distribution with mean e. Determine
the mean and the variance of the length of time until the
system fails.
4. Determine the mode of the gamma distribution with
parameters a and B.
5. Sketch the p.d.f of the exponential distribution for each
of the following values of the parameter 6: (a) B = 1/2, (b)
A = 1, and (c) 8 = 2.
6. Suppose that X1...., X, form a random sample of
size n from the exponential distribution with parameter
B. Determine the distribution of the sample mean X-
7. Let Xj. X, X be a random sample from the exponen-
tial distribution with parameter 8. Find the probability
that at least one of the random variables is greater than
1, where r>0.
11. Suppose that n items are being tested simultaneously,
the items are independent, and the length of life of each
item has the exponential distribution with parameter B.
Determine the expected length of time until three items
have failed. Hint: The required value is E(Y +Y2+ Y3) in
the notation of Theorem 5.7.11.
12. Consider again the electronic system described in Ex-
ercise 10, but suppose now that the system will continue
to operate until two components have failed. Determine
the mean and the variance of the length of time until the
8. Suppose that the random variables X1..... Xg are in-
dependent and X, has the exponential distribution with
parameter A, (i =1, ..., k). Let Y = min( X. .... X).
Show that Y has the exponential distribution with param-
eter f++ A.
system fails.
9. Suppose that a certain system contains three compo-
nents that function independently of each other and are
connected in series, as defined in Exercise 5 of Sec. 3.7,
so that the system fails as soon as one of the components
fails. Suppose that the length of life of the first compo-
13. Suppose that a certain examination is to be taken by
five students independently of one another, and the num-
ber of minutes required by any particular student to com-
plete the examination has the exponential distribution for
which the mean is 80. Suppose that the examination be-
gins at 9:00 A.M. Determine the probability that at least
one of the students will complete the examination before
9:40 AM.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F475f6844-e910-4a94-a368-6960b9a52137%2Fbc63e052-1456-4f0e-81fb-3cbd72e2791d%2F8t9qx6p_processed.jpeg&w=3840&q=75)
![](/static/compass_v2/shared-icons/check-mark.png)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
![A First Course in Probability (10th Edition)](https://www.bartleby.com/isbn_cover_images/9780134753119/9780134753119_smallCoverImage.gif)
![A First Course in Probability](https://www.bartleby.com/isbn_cover_images/9780321794772/9780321794772_smallCoverImage.gif)
![A First Course in Probability (10th Edition)](https://www.bartleby.com/isbn_cover_images/9780134753119/9780134753119_smallCoverImage.gif)
![A First Course in Probability](https://www.bartleby.com/isbn_cover_images/9780321794772/9780321794772_smallCoverImage.gif)