Derive the distribution of the range of a sample of size 2 from a distribut having density funct

Derive the distribution of the range of a sample of size 2...

Key Concepts

Change of Variables Technique

The change of variables technique is used to derive new distributions from existing ones by reparameterizing the variables. In this context, after expressing the joint distribution of the minimum and maximum in terms of new variables (such as the range and another auxiliary variable), one must compute the Jacobian of the transformation to correctly obtain the distribution of the range.

Joint Distribution of Order Statistics

For independent and identically distributed random variables, the joint distribution of order statistics, such as the minimum and maximum, is crucial for deriving related functions like the range. This joint density is obtained by considering both the probabilities associated with each order and the appropriate combinatorial factors, allowing for integration over the suitable region to find the marginal distribution of the range.

Transformation of Random Variables

Transforming random variables is a key technique used to derive the distribution of a function of random variables, such as the range. By applying change of variable methods, often involving Jacobians, one can convert the joint distribution of the original variables (or order statistics) to the distribution of their transformation (in this case, the difference between maximum and minimum).

Order Statistics

Order statistics are the sorted values from a sample of random variables. They are fundamental in statistical analysis for understanding properties like minima, maxima, medians, and other quantiles. In deriving the distribution of the range, the smallest and largest values (i.e. the first and last order statistics) are specifically used, as the range is defined by their difference.

Range of a Sample

The range of a sample is defined as the difference between the maximum and minimum values of that sample. Its distribution can be derived by using the joint distribution of the minimum and maximum order statistics. This concept is important as it provides a measure of dispersion in a data set, and determining its probability distribution requires careful manipulation of the variables' joint behavior.

Transcript

00:01 We are given the probability density function of a random variable x has 2x, 0 less than x less than 1 and 0 else.

00:10 Now we are actually picking up two random variables x1 and x2 i .d with this distribution.

00:18 All right.

00:20 We need to find the distribution of the range.

00:23 Range is actually defined as w is max of x1 comma x2 minus minof x1 comma x2 because range is highest minus lowest.

00:33 Because there are only two random variables.

00:35 So basically mod of x1 minus x2 is my random variable w and x1a next to are coming from this distribution.

00:45 So we need to find what is the probability density function of this w that is the question.

00:53 So let's start with cdf.

00:56 So what is the cdf definition? it is f w of w is probability that w is less than equal to w.

01:05 That is probability.

01:05 That is probability that mod of x1 minus x2 is less than equal to w.

01:11 Case 1, when w is negative, obviously the probability will be 0.

01:16 When w is negative, probability will be 0 because this is a positive number.

01:20 How can a positive number be less than a negative number? so it's an impossible event.

01:24 So that means fw of w is 0.

01:27 Now let's go to case 2.

01:29 Case 2 is what when w is greater than 0.

01:32 Equal to 0.

01:34 In that case, my f w of w is probability that modulus of x1 minus x2 is less than it equal to w, which i can write it as probability that minus w is less than it equal to x1 minus x2 is less than it equal to w.

01:52 Remember, w is a positive number now.

01:54 This x1 and x2 are random variables coming from which distribution.

01:58 It is f of x is equal to 2x, but both x1 and x2 are less than 0 less than x1 less than 1 and 0 less than x2 less than 1 so what we do is let's draw x1 axis and x2 axis so this is my x1 axis and this my x2 axis this is 0 this is 1 this is 1 this is 1 this is 1 basically the actual region is the square now we need to find the probability that x1 minus x2 lies between minus w and w w is a positive number all right so what is the graph of x1 minus x2 is equal to w x1 is equal to x2 plus w so this is of the form x minus y is equal to w or y is equal to x minus w so this is the graph and we need to shade the region x minus y is less than it equal to w now if w is greater than one if w is greater than one this straight line will be some something like this so i'll use a different color so this is a pink color straight line so w is greater than one and what about the other straight line the other straight line is this one right so it is like x minus y is equal to negative w so that straight line is also parallel to this pink color straight line it will go somewhere like this so this will be the straight line now what do you mean by minus w is less than it equal to x1 minus x2 less than it equal to w it is the region between these two straight lines obviously the entire square is in between this region so the probability is 1.

03:43 So, that means if w is greater than 1, f w of w is equal to 1.

03:49 So as of now, we got two conclusions...

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