Advice on Exponential, Binomial, & Normal Distributions

In summary, In the conversation, the individual is seeking help with solving probability questions related to various distributions such as Exponential, Normal, Binomial, and Poisson. They are unsure about their answers and are looking for clarification on certain aspects. They have managed to solve most of the questions except for one involving the Normal Distribution.
  • #1
NW8800
16
0
Hey,

I'm new to all this so cut me some slack, but have been trying to work through some questions, and I can't seem to find answers to these questions... Or atleast find the confidence that my answers/working is correct...


1. (Exponential Distribution) Telephone calls arrive at the information desk of a large
computer software company at the rate of 15 per hour.
What is the probability that the next call will arrive within 15 minutes?
(What would eb the formula for this one?) I got 0.98347


2. (Normal Distribution) For a group of trucks, it was found on an annual basis that the distance traveled per truck is normally distributed with a mean of 50.0 thousand km and a standard deviation of 12.0 thousand km.

(i) What proportion of trucks can be expected to travel between 34.0 and 38.0
thousand km in the year?


I get around 6.64%, but I am not sure that I am doing it right as I tried to do the same thing on the computer and I get something like 6.674438...

(ii) How many km will be traveled by at least 80% of the trucks?

(This one really has me stumpted as I am not sure if it means I look at the normal distro curve, and sort of 40% on each side or, do I try and figure out the 20% value there, and say that at least 80% of the trucks travel >than that distance. The latter makes more sense to me...) I tried the latter and I got 39,920km


3. (Binomial Distribution) A student is taking a multiple-choice exam in which each
question has four choices.


Assuming that he/she has no knowledge of the correct answers to any of the questions, he decided on a strategy in which he will place four balls (marked A, B, C and D) into a box.
He randomly selects one ball (with replacement)for each question. The marking on the ball will determine hisanswer to the question.
There are five multiple-choice questions on the exam. What is the probability the he will get:


(i) five questions correct?
(ii) at least four questions correct?


I think i have this one sorted now, i)=1/1024 ii)=5/1024


4. (Poisson Distribution) Assume that the number of network errors experienced in a day on
a local area network (LAN) is distributed as a Poisson random variable. The average
number of network errors experienced in a day is 2.4. What is the probability that in any
given day
(i) exactly one network error will occur?
(ii) two or more network errors will occur?


i) This one I get ~21.7%
ii) I did 1-P(X=2) = ~78%


Any help please?


Thanks,

NW
 
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  • #2
(ii) How many km will be traveled by at least 80% of the trucks?

(This one really has me stumpted as I am not sure if it means I look at the normal distro curve, and sort of 40% on each side or, do I try and figure out the 20% value there,
You are doing a 2-sided test, so you need 10% on each side.

For 4.(ii), 1-P(X=2) is the probability that X is not 2. Is that what you wanted?

It would help if you describe how you got each answer.
 
Last edited:
  • #3
Hey,

Thanks for your reply!

I seem to have it all sorted cept for question 2:

2. (Normal Distribution) For a group of trucks, it was found on an annual basis that the distance traveled per truck is normally distributed with a mean of 50.0 thousand km and a standard deviation of 12.0 thousand km.

(i) What proportion of trucks can be expected to travel between 34.0 and 38.0
thousand km in the year?

(ii) How many km will be traveled by at least 80% of the trucks?

Could you explain a little more about this one...
 
  • #4
(i) is the probability {34 < X < 38}. Do you know how to standardize the 34 and the 38 (turning them into a z score each)? Once you have that, you'd just need to look it up in a normal probability table.

(ii) is the answer to Prob{Z < z} = 0.8 solved for the little z. If F is the cumulative normal distribution so that F(z) = Prob{Z < z}, F(z) = 0.8 solved for z is your answer. Again, you can look up which value of z satisfies this from a normal probability table. Most probability tables assume a two-sided test, but you are running a one-sided test. So you need to look for the z value that leaves 0.2 probability "to the right." In a two-sided test, that z value will be stated as the z value corresponding to a 0.4 tail probability (0.2 to the left of -z, 0.2 to the right of +z). You are concerned only about the +z.
 

1. What is the difference between exponential, binomial, and normal distributions?

Exponential, binomial, and normal distributions are all types of probability distributions used in statistics. The main difference between them is the shape of the curve. Exponential distributions are used to model the time between events occurring, binomial distributions are used to model the number of successes in a fixed number of trials, and normal distributions are used to model continuous data that follows a bell-shaped curve.

2. How are these distributions used in real life?

Exponential, binomial, and normal distributions are used in a variety of fields, such as finance, biology, psychology, and engineering. They can be used to analyze data and make predictions about future events or outcomes.

3. What are the parameters of these distributions?

The parameters of a distribution are the values that determine the shape and characteristics of the distribution. For exponential distributions, the parameter is the rate of decay. For binomial distributions, the parameters are the number of trials and the probability of success. For normal distributions, the parameters are the mean and standard deviation.

4. How do you interpret the results of these distributions?

The results of these distributions can be interpreted in terms of probabilities. For example, in a binomial distribution, the probability of getting a certain number of successes can be calculated. In a normal distribution, the probability of a data point falling within a certain range can be calculated. These probabilities can be used to make predictions and inform decision making.

5. Can you give an example of a real-life application of these distributions?

One example of a real-life application of these distributions is in risk management. For instance, an insurance company may use exponential distributions to model the time between accidents, binomial distributions to model the number of insurance claims in a given time period, and normal distributions to model the cost of claims. This information can help the company assess and manage their risk effectively.

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