Find Expected Value of X(n) from Uniform Distribution

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In summary, the conversation discusses finding the Expected Value of the largest observation in a sample from a uniform distribution. To do so, one must first find the distribution of the maximum observation and then differentiate it to find the density. Finally, the expected value can be calculated by integrating the density function.
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apalmer3
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I swear that I used to know this.

If you have an independent sample of size n, from the uniform distribution (interval [0,[tex]\theta[/tex]]), how do you find the Expected Value of the largest observation(X(n))?
 
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If [tex] X_{(n)}[/tex] is the maximum in the sample, you first find its distribution. Since you have a random sample of size [tex] n [/tex], you can write

[tex]
F(t) = \Pr(X_{(n)} \le t) = \prod_{i=1}^n \Pr(X_i \le t) = \left(\frac{t}{\theta}\right)^n
[/tex]

Differentiate this w.r.t. [tex] t [/tex] to find the density [tex] f(t) [/tex], and the expected value is

[tex]
\int_0^{\theta} t f(t) \, dt
[/tex]
 

FAQ: Find Expected Value of X(n) from Uniform Distribution

What is the Uniform Distribution?

The Uniform Distribution is a probability distribution that assigns equal probability to all possible outcomes within a given range. In other words, it is a continuous distribution where all values within a specific range have the same likelihood of occurring.

How is the Expected Value of X(n) from Uniform Distribution calculated?

The Expected Value of X(n) from Uniform Distribution is calculated by taking the average of all possible outcomes within a given range. This is done by multiplying each possible outcome by its corresponding probability, and then summing up all these values.

What is the significance of finding the Expected Value of X(n) from Uniform Distribution?

The Expected Value of X(n) from Uniform Distribution is important because it provides a measure of central tendency for the distribution. It can be used to make predictions about future outcomes and can also help in decision-making processes.

Can the Expected Value of X(n) from Uniform Distribution be negative?

No, the Expected Value of X(n) from Uniform Distribution cannot be negative. This is because the Uniform Distribution assigns equal probability to all outcomes within a given range, and therefore the average of these values will always be positive or zero.

How does the range of values in the Uniform Distribution affect the Expected Value of X(n)?

The range of values in the Uniform Distribution does not affect the Expected Value of X(n). This is because the distribution is characterized by equal probabilities for all possible outcomes, regardless of the range. The Expected Value will only change if the probabilities assigned to each outcome are different.

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