Grewal

Question

Suppose avg waiting time of a process to get chance in a queue is 5 min. What will the probability that process get chance at first minute is ________________

Answer 1

For the detailed solution , please see the pic below

Comments

@srestha mam, please see the pic below for the answer.

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Here the lifetime of the product is 0<T<3

T>=2 states that the product will work fine till 2 or more years i.e it includes 0<T<=2 and 2<T<3 and T>=3. 

But as the product breaks down in 3rd year, we have to remove T>=3 from above. 

Thus the required probability is P(T>=2) - P(T>=3).

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Probability will be $\frac{e^{-\frac{2}{4}}-e^{-\frac{3}{4}}}{e^{-\frac{2}{4}}}$

right??

i.e. $P\left ( A/B \right )=\frac{P\left ( A\cap B \right )}{P\left ( B \right )}$

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No.

Required probability= P(0<T<3) = P(T>=2)-P(T>=3)

I had given the explanation.

Answer 2

Avg waiting time $\lambda$ = $5$ min.

average rate is $1$ process in $5$ minutes i.e. $1/5$

According to exponential distribution,

$P(X<=1) = \int_{0}^{1}\frac{1}{5}e^{-\frac{t}{5}}dt = 1- e^{-\frac{1}{5}} = 1- 0.81=0.18$

Comments

I think it will be $e^{-{5}}$

chk it https://courses.lumenlearning.com/introstats1/chapter/the-exponential-distribution/

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But in every question it is mentioned that it is exponentially distributed. Also in the link check the relation between exponential and poisson distribution.

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with exponential distribution, how much do u get?

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average rate is 1 process in 5 minutes i.e. 1/5

P(X<=1) = $\int_{0}^{1}\frac{1}{5}e^{-\frac{t}{5}}dt = 1- e^{-\frac{1}{5}} = 1- 0.81=0.18$

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Here we need to find $P\left ( X=1 \right )$, less than sign not required.

What will be $\mu ?$ Is it $5$ or $\frac{1}{5}?$

Here exponential distribution is more appropriate than poisson distribution. right?

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why we  don't need to find before 1st minute ? at first minute means from 0 to 60 sec right ? we will just calculate pdf not cdf ?

mean will be 1/ $\lambda$ = 5.

Yes exponential will be more appropriate.

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@Satbir

but why r u converting it in second?

Everything is minute here

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I am using minute only. Just want to say that it is continous and not discrete thats why we need to use cdf and not pdf right ?

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where do u got pdf is discrete case?

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Another point, why exponential better here?

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CDF is cumulative so it will include past values also right ? but Pdf will only calculate at P(x=1) i.e at a particular point.

 

In the question probability of waiting time is asked.

The waiting times for poisson distribution is an exponential distribution with parameter lambda.

https://stats.stackexchange.com/questions/2092/relationship-between-poisson-and-exponential-distribution

Read the 2nd last answer.

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Execution of process starts from t=0 or 0th time instant( here minute ). In the question it is mentioned to find the probability that the process gets chance to execute at the first minute i.e. t=1.

If the question says average service time then exponential distribution should be used.

If the question says numbers of events occured or arrived at a given time period, then poisson distribution is used.

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@SuvasishDutta

So here we will use poisson or exponential ? and PDF will be used or CDF ?

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Here we will use exponential distribution.

Between cdf and pdf which one to use is shown in the pic below. 

In the pic, it is explained that if we use either cdf or pdf the value will remain unchanged. Hence for this question any of them can be used.

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So it doesn't matter whether the given values we take them as discrete values or continous values ?

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No it matters.

In the previous comment i am saying that the value of cdf and pdf or value of cdf and pmf are same.

Discrete or continuous values depends on what distribution we use.

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I have provided an answer to the question. Please check.

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So, the conclusion is

first we see what distribution to use by looking whether we have to calculate it for discrete values or continous and other things like what is asked in question.

then we can either calculate its pdf or cdf

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Absolutely perfect @Satbir .

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@Satbir

pdf and cdf both can take continuous value

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okay

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@srestha mam,

pdf can take only continuous values,

pmf can take only discrete values,

but cdf can take both discrete and continuous values.

For continuous probability distribution, c.d.f take continuous values i.e. a range of values. 

For discrete probability distribution, c.d.f take discrete values.

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@SuvasishDutta

yes, right.

By the way, from where u read for  this probability portion, like exponential, pmf, pdf, cdf portion?

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From grewal book and made easy engg maths book

Answer 3

By Using the poisson distribution and substituting the average rate to 3,

Probality is 0.149

Comments

not clear