GATE CSE 2011 | Question: 33

Question

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?

• A. Index position of mode of $X$ in $X$ is the same as the index position of mode of $Y$ in $Y$
• B. Index position of median of $X$ in $X$ is the same as the index position of median of $Y$ in $Y$
• C. $\mu_y=a \mu_x + b$
• D. $\sigma_y=a \sigma_x + b$

Comments

Effects of Linear Transformations on mean and standard deviation

http://onlinestatbook.com/2/summarizing_distributions/transformations.html

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

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Ans by @Arjun

 

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Source link

Answer 1

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.

Comments

Sanjeev kumar Sen standard deviation is square root of variance….

and it is deviation from mean...in sir’s example

it should be divided by n-1

SD of X = sqrt((16.6^2 + 13.6^2 + 16.4^2 + 14.4^2 + 0.6^2)/4) = 15.306

SD of Y = sqrt((33.2^2 + 27.2^2 + 32.8^2 + 28.8^2 + 1.2^2)/4) = 30.613 = 2 * SD of X. 

 

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Do this condition true for divsion and subtraction too

 

0

 

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@Arjun "In (D) choice, if "+b" is removed, will it be correct?"

No, Correct statement shoud be
$\sigma_y = \sqrt{\sigma^{2}_{y}} = \sqrt{Var[aX+b]} = \sqrt{a^{2}Var[X]} = \lvert a\rvert \sigma_x $

Answer 2

(D) Answer.  σy=aσx

Comments

sir , can you explain how did changed value of E[Y] = uy for standard deviation case

Answer 3

Taking example here also give ansing easy

Say

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

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 $a\sigma _{x}+b=2.28$

https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/mean-median-and-mode-review

https://en.wikipedia.org/wiki/Standard_deviation

Comments

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

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@eyeamgj why??

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Another formula:

For symmetry ditribution

• mean-mode=3(mean-median)

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• mean-mode=3(mean-median)

 Only for moderately skewed distribution.

Answer 4

Let Variance of X = p

$\sigma _{x}$ = $\sqrt{p}$

Variance of Y = $a^{2}p$ 

 ($\because$ 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. when we multiply a random variable with a constant, since we square the values while calculating variance, the value of a(constant) will also gets squared)

$\sigma _{y}$ = $a\sqrt{p}$ = a * $\sigma _{x}$

$\therefore $ $\sigma _{y}$ =  $a \sigma _{x}$

Answer D is incorrect

 

Comments

Do this condition true for divsion and subtraction too

Answer 5

Here is the reason why option D is wrong : 

Answer 6

Simple proof for Option D:   Let Y = aX + b.

We know:  Var(Y) = E(Y²) − [E(Y)]²

Now, E(Y) = aE(X) + b

and  E(Y²) = E[(aX+b)²] = a²E(X²) + 2abE(X) + b²

Therefore,

Var(Y) = a²E(X²) + 2abE(X) + b² − [aE(X)+b]²

           = a²[E(X²) − (E(X))²]  = a²Var(X)

Hence,

σY = |a|σX

Since a > 0,  σY = aσX

So, the +b term does not appear in standard deviation.

Therefore, if Option D says σY = aσX + b, it is incorrect.

Reference: ChatGPT