GATE CSE 2021 | Set 2 | Question: 22

Question

For a given biased coin, the probability that the outcome of a toss is a head is $0.4$. This coin is tossed $1,000$ times. Let $X$ denote the random variable whose value is the number of times that head appeared in these $1,000$ tosses. The standard deviation of $X$ (rounded to $2$ decimal place) is _________

Comments

Here, X is a binomial random variable. For X, variance is n*p*(1-p).
Here, p is probability of getting a heads i.e. p = 0.4.
So, variance = 1000 * 0.4 * 0.6
Hence, standard deviation = square root of variance = 15.49 (approx to two decimal places)

Answer 1

It is Independent trails with same rate of success. So we can use Binomial distribution

Probability of success $(p) = 0.4$

Probability of failure $(q) = 1 – 0.4 = 0.6$

Number of trials $(n) = 1000$

Variance $=n\times p\times q = 1000 * 0.4 * 0.6 = 240$

Standard deviation $=\sqrt{240} = 15.49$

Comments

Why we don't use Poisson distribution?

λ = np = 1000 * 0.4 = 400

In Poisson, mean = variance = λ

So variance = 400 and standard deviation = √400 = 20

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@Santi Singha
Normally, Poisson distribution ka use more suitable hota hai jab p kafi chhota hota hai (jaise 0.01 ya usse kam).

Binomial distribution ka use tab hota hai jab:

• Fixed number of trial hote hai.
• Har trial me sirf do outcomes possible hote hain (success ya failure).
• Har trial independent hota hai.
• Success ki probability p har trial me same hota hai.

Poisson Distribution

Poisson distribution typically use hota hai jab:

• Keywords dikhe : jaise accidents per day, calls per hour, etc.
• Ye "rare events" ke liye use kiya jata hai jab n bahut bada hota hai aur p bahut chhota hota hai.

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we cant because in poissons the probab. of success is very very small

 

Answer 2

Good Luck