Working out the cumulative distribution

  • Thread starter millwallcrazy
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In summary, the cumulative distribution for the largest item in a random sample of size n is equal to the probability that all items in the sample are less than or equal to a, where a is the maximum value. This is based on the independence of the X values in the sample.
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millwallcrazy
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f(x) = 3(x^2)/(C^3) 0 < x < C
= 0 otherwiseLet the mean of the sample be Xa and let the largest item in the sample be Xm. What is the cumulative distribution for Xm?
 
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  • #2
I'm assuming that you are to work with a random sample of size [itex] n [/itex].

Note that for ANY continuous random variable, if [itex] X_{max} [/itex] is the maximum value, you know that [itex] X_{max} \le a [/itex] means that EVERY item in the sample is [itex] \le a [/itex], so that

[tex]
G(a) = P(X_{max} \le a) = P(X_1 \le a \text{ and } X_2 \le a \text{ and } \dots \text{ and } X_n \le a)
[/tex]

Now, knowing that the [itex] X [/itex] values are independent (since they're from a random sample), what can you do with the statement on the right?
 

1. What is the purpose of working out the cumulative distribution?

The cumulative distribution is used to show the probability of a random variable being less than or equal to a certain value. It is helpful in understanding the likelihood of a certain event occurring.

2. How is the cumulative distribution calculated?

The cumulative distribution is calculated by adding up the probabilities of all values up to and including the desired value. This can be done manually or by using statistical software.

3. What is the difference between the cumulative distribution and the probability distribution?

The probability distribution shows the probabilities of all possible values of a random variable, while the cumulative distribution shows the probability of a value being less than or equal to a certain value.

4. Can the cumulative distribution be used for continuous data?

Yes, the cumulative distribution can be used for both discrete and continuous data. However, for continuous data, the cumulative distribution is represented by a continuous curve instead of a step function.

5. How can the cumulative distribution be interpreted?

The cumulative distribution can be interpreted as the area under the probability distribution curve up to a certain value. It can also be interpreted as the probability of a random variable being less than or equal to a certain value.

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