GATE CSE 2017 Set 1 | Question: 19

Question

Let $X$ be a Gaussian random variable with mean 0 and variance $\sigma ^{2}$. Let $Y$ = $\max\left ( X,0 \right )$ where $\max\left ( a,b \right )$ is the maximum of $a$ and $b$. The median of $Y$ is ______________ .

Comments

@jiminpark 

1. JBStatistics videos on youtube is a good start.
2. yes, Normal distribution is also kmown as gaussian distribution.

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@abcS Thank you Sir !

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@jiminpark I’m not a sir. An aspirant, just like you. :)

Answer 1

For Gaussian distribution, Mean=MEDIAN=Mode, hence for the random variable X, Median = Mean=0. This also means that if we arrange all the real values taken by X with repetition in increasing order, then the sequence would look like

……….-3,-3,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0(Median),0,0,0,0,1,1,1,1,1,2,2,2,3,3…...{Sequence is not exactly scaled according to the Gaussian Distribution, but it roughly shows how listing the whole distribution would look like. Also 0 being the mode, I have listed it comparatively more no. of times.}

Now, random variable Y=max(X,0)={0,for Y<=0

                                                                X,for Y>0}

So if we list distribution of Y, it would look like-

………..0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(Median),0,0,0,0,1,1,1,1,1,2,2,2,3,3…….

since median is position dependent, so 0 remains the Median since only the entries smaller than it are replaced by 0 itself, without affecting its position. Also note that Mode of Y is 0 too. But mean of Y will shift away from 0 in the positive direction, as the distribution has become skewed.