Working out the cumulative distribution

millwallcrazy · May 8, 2010

f(x) = 3(x^2)/(C^3) 0 < x < C
= 0 otherwise

Let the mean of the sample be Xa and let the largest item in the sample be Xm. What is the cumulative distribution for Xm?

statdad · May 8, 2010

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?

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Working out the cumulative distribution

millwallcrazy

statdad