GATE2012-21

Question

Consider a random variable $X$ that takes values $+1$ and $−1$ with probability $0.5$ each. The values of the cumulative distribution function $F(x)$ at $x = −1$ and $+1$ are

• A. $0$ and $0.5$
• B. $0$ and $1$
• C. $0.5$ and $1$
• D. $0.25$ and $0.75$

Answer 1

Given $P(-1) = 0.5$ and $P(1) = 0.5$. So, at all other points P must be zero as the sum of all probabilities must be 1.

$So, F(-1) = 0.5$ and

$F(1) = P(-1) + 0 + 0 + ... + P(1)$

= $0.5 + 0.5 = 1$

Correct Answer: $C$

Comments

@Arjun Sir as value of the random variables are only +1 and -1, So I thinks we should not consider the other points between -1 and +1. right ?

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It is not told that random variables take only 2 values. But instead told that probability of 2 values sum to 1. So, probability of other points must be 0. Even if we consider 2 points only, we get the answer.

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what is the formula of F(x)?

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It is the sum of all probability distribution values till the given point $x$. Use integration for continuous probability and summation for discrete. As the probability distribution function changes (like for uniform, poisson etc.) this formula also changes. This question does not require any formula or there is no use even if one had known all such formula. (One reason why I say formula are not important for GATE).

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@arjun sir,can you suggest a good site for knowledge of probability distribution functions and ol

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All good links I find are updated in respective subject pages in gatecse under weblinks- you can see here:

http://gatecse.in/probability/

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@arjun Sir....For F(-1) we have to consider all the points from neg infinity to -1 and same for F(1)?

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Yes, as per definition.

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F(1) = P(-1) + 0 + 0 + ... + P(1)

= 0.5 + 0.5 = 1

why so?? I am unable to understand. Plz explain :(

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That is completely theoretical. Intuitively we can say that sum of probabilities across all possible values is always 1. So if probability at two points are 0.5 each, then at all other points it must be 0. Now cumulative probability is nothing but sum of probabilities of all points till the given one.

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Thanx arjun sir :)

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Is it in GATE 2017 syllabus?

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Why not?

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In syllabus only 5 distributions are mentioned.

Binomial

Poissons

Uniform

Exponential

Normal

Is cumulative dis. is subcategory of any of these?

I never read about cumulative dis. So I am asking you...

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You are right. Officially it shouldn't be asked but it's just a 5 min thing. But yeah it's not in syllabus.

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It is in syllabus. The 5 given in syllabus are probability distribution functions and for each of them we have the cumulative distribution function.

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Sir IN the GATECSE PROBABILITY  2 links that are FORMULA FOR DISTRIBUTIONS AND NOTES FOR DISTRIBUTIONS ARE NOT opening.

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probability distribution

http://web.archive.org/web/20151010091326/http://web.stanford.edu:80/~iwright/PP%20Materials/Lesson%206%20Probability%20Distributions%20Notes.pdf

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Sir I m  confused about f(x) at c -1 is .5 why????  @Arjunsir