The maximum and minimum of a random sample of numbers are...

The maximum and minimum of a random sample of numbers are called the "extrema" of the sample. Fix positive integers $n$ and $N$. Suppose $n$ draws are made at random with replacement from the numbers \{1, 2, 3, ..., $N$\}. Let $X_i$ be the number that appears on the $i$th draw. Let $V_n = \min\{X_1, X_2, ..., X_n\}$ be the sample minimum and let $W_n = \max\{X_1, X_2, ..., X_n\}$ be the sample maximum. We will use simulations to compute an approximation to the distribution of $W_n$. (a) Maximum of draws Write a function named \texttt{max\_sim} that takes $N$ and $n$ as its arguments, randomly draws $n$ times with replacement from \{1, 2, 3, ..., $N$\} and returns the maximum of the draws.

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00:01 Hi, here it has been given that y n denote the maximum of a random sample of size n from a distribution of the continuous type that has cumulative distribution function fx and pdf x xx, fx, is equal to capital f -dash x and we have to find the limiting distribution of zn which has been given.

00:20 Okay, so let x1 x2 xn follows a distribution with probability density function fx and these random samples are iid and it has cumulative distribution function ctf fx and also it has given that small fx that is the probability density function is f -dash -x that means first order derivative of capital f -dash -x with respect to x and yn is the maximum order statistic of maximum of all the random samples that means maximum order statistics of all the random samples equals to y less than equals to n and it has been given that zn equals to n into one minus f y n we have to find the limiting distribution of this z n okay so let us take the cumulative distribution function cdf of z n b f n x where f n x x is is equal to probability z n is less than equals to small x now we can write this probability is equals to probability n into 1 minus f y n is less than equals to small x now this we can write 1 minus f y n is less than equals to x by n so from here we write probability f y n is greater than equals to 1 minus x by n right so from here we get probability y n is greater than equal to f inverse 1 minus x by n okay now see the maximum water statistics is greater than some quantity okay so from here we can write probability y n is less than f inverse 1 minus x by n right now see the maximum order statistic is less than some quantity right so when we have order statistics x order 1 that is much that must be less than x order 2 less than x order 3 and up to less than x order n now see this quantity x order n nth order statistic, which is also have denoted by yn, this quantity is less than some quantity.

03:04 That means all the random samples is less than this quantity, right? because the maximum is less than this quantity.

03:11 So, we can say all the random samples is less than this quantity.

03:14 So we can write 1 minus probability x1 is less than f inverse 1 minus x by n...

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