I Max of 3 random cards from deck vs max of 3 numbers from 1-13

[member 428835]

Hi PF!

I am wondering the differences between the discrete and continuous case for expected value of minimum of 3 integers uniformly distributed from 1 to 13 vs 3 reals from 1 to 13.

The real case is direct: $F = ((x-1)/12)^3 \implies f = 3(x-1)/12)^2$ for CDF $F$ and PDF $f$. Thus the expected value for the max of 3 reals in this range is $\int_1^{13} x f \, dx = 10$. But now for the discrete case: the probability a random variable $X_i$ is less than some integer $k$ I think should be $P(X_i \geq k) = (13-k+1)/13 \implies P(X \geq k) = ((13-k+1)/13)^3$ but I really don't know how to proceed. Is there a direct way to arriving at the CDF that I'm missing?

 

[PeroK]

Why not just $c(k) = (\frac k {13})^3$? And $p(k) = (\frac k {13})^3 - (\frac{k-1}{13})^3$?

 

[Office_Shredder]

Just to point out, drawing three cards from a deck is not the same as picking three integers uniformly at random, unless you replace the cards each time you draw.

Also. $P(X\geq k)$ is the cdf, well one minus that is. Were you not sure how to get the pdf (which as Perok points out is just the difference of consecutive pdfs).Note $k^3-(k-1)^3$ is actually a quadratic polynomial, so the answers are more similar than they might initially appear

 

[PeroK]

I got $E = \frac{133}{13} \approx 10$ for the discrete case.

 

[PeroK]

In general, for a uniform choice from $1-n$ (with replacement) of $m$ cards, then I get the expected value of the highest card to be:
$$E = n - \frac{1}{n^m}\sum_{k =1}^{n-1}k^m$$With $m = 3$, we have the sum of cubes:
$$\sum_{k =1}^{n-1}k^3 = \frac{n^2(n-1)^2}{4}$$And$$E = n - \frac{(n-1)^2}{4n}$$And with $n = 13$:
$$E = 13 - \frac{36}{13} = \frac{133}{13}$$

 

[Mark44]

@joshmccraney, your thread title is misleading: "max of 3 random cards from deck vs max of 3 numbers from 1-13"

A deck of cards has 52 cards in it, in four suits. Unless your deck has just 13 cards -- A, 2, 3, ..., J, Q, K -- in one suit, it's different from the set of integers 1 through 13.

The first post seems to be asking a different question -- the difference between a discrete set and a continuous set.