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Crank
3 years ago
14

The time between arrivals of customers at an automatic teller machine is an exponential random variable with a mean of 4 minutes

. Round yours answers to 4 decimal places. (a) What is the probability that more than three customers arrive in 10 minutes
Mathematics
1 answer:
andre [41]3 years ago
6 0

Answer:

0.7788 = 77.88% probability that more than three customers arrive in 10 minutes

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

f(x) = \mu e^{-\mu x}

In which \mu = \frac{1}{m} is the decay parameter.

The probability that x is lower or equal to a is given by:

P(X \leq x) = \int\limits^a_0 {f(x)} \, dx

Which has the following solution:

P(X \leq x) = 1 - e^{-\mu x}

The probability of finding a value higher than x is:

P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}

In this question, we have that:

The time between arrivals of customers at an automatic teller machine is an exponential random variable with a mean of 4 minutes, and we have three customers, which means that m = 3*4 = 12, \mu = \frac{1}{12} = 0.0833

(a) What is the probability that more than three customers arrive in 10 minutes

This is P(X > 3). So

P(X > 3) = e^{-0.0833*3} = 0.7788

0.7788 = 77.88% probability that more than three customers arrive in 10 minutes

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Answer:

absolute max: f(x,y)=1/2+p1 ; at P(1/2,1/2)

absolute min: f(x,y)=p1 ; at U(0,0), V(1,0) and W(0,1)

Step-by-step explanation:

In order to find the absolute max and min, we need to analyse the region inside the quarter disc and the region at the limit of the disc:

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There could be Minimums and Maximums, if:

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(1-2x, 1-2y)=(0,0)

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f(P)=1/2+1/2+p1-1/4-1/4=1/2+p1

f(0,0)=p1

We see that:

f(P)>f(0,0), then P(1/2,1/2) is a maximum relative

<u>Region at the limit of the disc:</u>

We use the Method of Lagrange Multipliers, when we need to find a max o min from a f(x,y) subject to a constraint g(x,y); g(x,y)=K (constant). In our case the constraint are the curves of the quarter disc:

g1(x, y)=x^2+y^2=1

g2(x, y)=x=0

g3(x, y)=y=0

We can obtain the critical points (maximums and minimums) subject to the constraint by solving the system of equations:

∇f(x,y)=λ∇g(x,y) ; (gradient)

g(x,y)=K

<u>Analyse in g2:</u>

x=0;

1-2y=0;

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Q(0,1/2) critical point

f(Q)=1/4+p1

We do the same reflexion as for P. Q is a maximum relative

<u>Analyse in g3:</u>

y=0;

1-2x=0;

x=1/2

R(1/2,0) critical point

f(R)=1/4+p1

We do the same reflexion as for P. R is a maximum relative

<u>Analyse in g1:</u>

(1-2x, 1-2y)=λ(2x,2y)

x^2+y^2=1

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For \lambda_{1}: S(x,y)=(0.70, 070)

and

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we need also to analyse the points limits between g1, g2 y g3, that means U(0,0), V(1,0), W(0,1)

f(U)=p1

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absolute max: f(x,y)=1/2+p1 ; at P(1/2,1/2)

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