A median value of any probability distribution divides the area under the probabaility distribution in two equal parts.
Best understood with following image.
![Image](https://gre.myprepclub.com/forum/download/file.php?mode=view&id=927)
Now in our case we need to split the triangular distribution along a line \(x=?\) (parallel to y axis) such that area of the right half is same as the left half.
![Image](https://gre.myprepclub.com/forum/download/file.php?mode=view&id=928)
Area of right half = \(\frac{1}{2} \times\) Area of the larger triangle.
Now look at the figure below and we have marked out the hight and length of the triangle as x. Now height = length for this triangle because the given line has slope 1.
![Image](https://gre.myprepclub.com/forum/download/file.php?mode=view&id=928)
Area of right half = \(\frac{1}{2} \times x^2\)= \(\frac{1}{2} \times \frac{1}{2}\times \sqrt{2}^2\).
Solving for x we get x =1. So median value has to be \(\sqrt{2}-1\) (refer to the figure above)
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