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+20 votes

Consider a finite sequence of random values $X=[x_1,x_2,\dots x_n]$. Let $\mu_x$ be the mean and $\sigma_x$ be the standard deviation of $X$. Let another finite sequence $Y$ of equal length be derived from this as $y_i=a*x_i+b$, where $a$ and $b$ are positive constants. Let $\mu_y$ be the mean and $\sigma_y$ be the standard deviation of this sequence.

Which one of the following statements is INCORRECT?

  1. Index position of mode of $X$ in $X$ is the same as the index position of mode of $Y$ in $Y$
  2. Index position of median of $X$ in $X$ is the same as the index position of median of $Y$ in $Y$
  3. $\mu_y=a \mu_x + b$
  4. $\sigma_y=a \sigma_x + b$
asked in Probability by Veteran (115k points)
edited by | 2.1k views

This might help ...


Effects of Linear Transformations on mean and standard deviation


Practice question of similar type but with normal distribution rather than discrete. 

3 Answers

+25 votes
Best answer
Answer - $D$.

Mean, median and mode are linear functions over a random vaiable.

So, multiplying by constants or adding constants wont change their relative position.

Standard deviation is not a linear function over a random variable.
answered by Loyal (8.8k points)
edited by
In (D) choice, if "+b" is removed, will it be correct?

i dont think so

since sigma2(x) = E(x2) - E(x)2 sigma(y) will be dependent upon b since it is part of E(y)


Okay, lets try an example:

Let X = { 1, 4, 34, 32, 17}
Mean  = 88/5 = 17.6
SD = sqrt(16.62 + 13.62 + 16.42 + 14.42 + 0.62) = 30.61

Now, I create Y using a = 2 and b = 3. 
Y = {5, 11, 71, 67, 37}
Mean = 38.2
SD = sqrt(33.2+ 27.22 + 32.82 + 28.82 + 1.22) = 61.22 = 2 * SD of X. 

1 example is not enough to prove this. But this statement would be enough:

When we add a constant to each element of the list, the standard deviation (or variance) remains unchanged. This is because, the mean also gets added by the same constant and hence the deviation from the mean remains the same for each element.

makes sense
sir, in that statement it says "when we add a constant to each element of the list, the standard deviation remains unchanged." but in your eg solution itselt, SD(Y)=2*SD(X). Please explain this.
I think apart from adding 3, we are also multiplying each term with 2, that's why s.d getting doubled.If we only add something then, s.d will remain constant.
yes i also think when we add only then only sd will remain constant here
For Random  Variable X(Discrete here)

Mean = Expected value(X) = E(X) =  $\sum p_{X}\left ( x \right )*x$

Variance(X) = $E\left [ X^{2} \right ] - [E\left [ X \right ]]^{2}$

What about Median and Mode ????


i think in your example, 

$\sigma_x$  will be 13.69

$\sigma_y$  will be 27.25


+12 votes

(D) Answer.  σy=aσx

answered by Active (2.4k points)
+3 votes

Taking example here also give ansing easy


$X= \left [ 2,3,4,4,5 \right ]$

and value of $a=2 , b=100$

So, according to question $Y= \left [ 104,106,108,108,110 \right ]$

Now come to option

A) correct, because for X mode is 4 and for Y mode is 108 which has same index position

B) correct , because for X median is 4and for Y median is 108 which has same index position

C) both gives mean =107.2

D) will be incorrect in option elimination

and also by calculation

$\sigma _{y}=0$

while $a\sigma _{x}+b=2.28$

answered by Veteran (111k points)
edited by

srestha MAM  I THINK  aσx+b=2.28 IS WRONGLY CALCULATED??


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