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## Two brands A and B of a product have probabilities 30% and 70% respectively at time t=0, if their transition matrix be

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$\left[\begin{array}{cc}.7& .3\\ .2& .8\end{array}\right]$, Find their probabilities.

1. after t=1
2. After t=2

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## A supermarket has two girls running up sales at the counters. If the service time for each customer is exponentially distributed with mean of 4 minutes and if the arrival poissonian with mean rate of 10/hr, then find.

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1. The probability of having to wait for service,
2. The expected idel time for each girl

## Prove that the probability distribution in Poission queues is given by :

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${P}_{n}\left(t\right)={e}^{-\lambda t}\frac{\left(\lambda t{\right)}^{n}}{⌊n⌋}$ Where t denotes number of arrivals n in a time interval t and λ denotes average rate of arrivals.

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## Obtain regression line of x on y for the given data:

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 x 1 2 3 4 5 6 y 5 8.1 10.6 13.1 16.2 20

## Fit the second degree parabola to the following data taking x as the independent variable.

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 x 1 2 3 4 5 6 7 8 9 y 2 6 7 8 10 11 11 10 9

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n=18,

## Calculate the coefficient of correlation between x and y from the following data.

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 x -10 -5 0 5 10 y 5 9 7 11 13

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## The joint probability mass function of (x,y) is given by P(X=x,Y=y)=k(2x+3y), X=0,1,2: Y = 1,2,3. Find k, marginal probability distribution of X and Y. Also find conditional probability distribution of X for Y=1. i.e.

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$P\left[\frac{X=x}{Y=1}\right]$

## For the binomial variate prove the recurrence formula is

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${\mu }_{k+1}=pq\left[\frac{d{\mu }_{k}}{dp}+nk{\mu }_{k-1}\right]$

Where μk is Kth order central moment.

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