Deriving a Maximum’s CDF Under Independence

Summary

The document explains how to derive the cumulative distribution function of the maximum of several random variables when they are independent. To find the probability that the maximum is at most a threshold, it first considers the complementary event that all variables exceed that threshold. Independence allows the joint probability to be written as a product of the individual probabilities; for uniform variables on the unit interval, this yields the stated CDF, 1 − (1 − m)^n.

The key caveat is that the product step depends on independence. Without that assumption, the marginal distributions alone do not determine the probability of the joint event; additional information about the variables’ joint distribution is needed. The response gives a concise derivation but does not discuss dependent cases or other distributions.

Key ideas

• The maximum is at most a threshold when every variable is at most that threshold.
• The complementary event occurs when all variables exceed the threshold.
• Independence lets the joint probability factor into a product of marginal probabilities.
• Without independence, the joint distribution is needed to determine the maximum’s CDF.

Tags

# How to derive the CDF and the probability density function


# How to derive the CDF and the probability density function












Is there something missing in this question i dont seem to understand, can anyone help explaining what is required?

## Answer by vagoum (score 3)

https://quant.stackexchange.com/a/50128

During the calculation of the distribution function of $M$, that is $ P(M \leq m)$, there is an independency assumption being used. That is the condition you are missing, it seems like it was forgotten.

$ P(M \geq m) = P(X_1 \geq m, X_2 \geq m, ... X_n \geq m) = $ (missing the independency condition here)

$P(X_1 \geq m)P(X_2 \geq m)...P(X_n \geq m)= (1-m)^n$

So $P(M \leq m) = 1-(1-m)^n$, as usual.

Without the independency condition you cannot proceed further in the calculation, unless you know more about the joint distribution of the $X_i$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.