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Math Help - Homework - Probability Density Function

  1. #1
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    At my local sandwich bar, I have noticed that it always takes at least two minutes to serve a customer and it can take much longer to fulfil an unusual order. The time in minutes taken to serve a customer may be modelled by a continuous random variable T with probability density function f(t)= 64 t5 ,t≥ 2.
    (i) Show that the c.d.f. of the random variable T is given by
    F(t)= 1 − 16 t4 ,t≥ 2.

    (ii) According to the model, what proportion of customers take more than four
    minutes to serve?
    Find the probability that it takes between five and ten minutes to serve a
    customer.

    (iii) Use the p.d.f. f(t) to calculate the mean and variance of the time taken to serve a customer.

    (iv) Use Formula (4.1) of Unit 1 to calculate the mean time taken to serve a
    custome.
    Formula (4.1) = (see below)
    (v) Simulate the time taken to serve a customer, using the random number
    u=0.5536, which is an observation from the uniform distribution U(0,1).
    Give your answer in minutes and seconds, to the nearest second.

    foruma attached.
    Attached Thumbnails Attached Thumbnails Homework - Probability Density Function-test2.png  
    Last edited by mr fantastic; March 8th 2009 at 05:05 AM. Reason: Merged posts, editing
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  2. #2
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    Quote Originally Posted by wallace View Post
    At my local sandwich bar, I have noticed that it always takes at least two minutes to serve a customer and it can take much longer to fulfil an unusual order. The time in minutes taken to serve a customer may be modelled by a continuous random variable T with probability density function f(t)= 64 t5 ,t≥ 2.

    (i) Show that the c.d.f. of the random variable T is given by

    F(t)= 1 − 16 t4 ,t≥ 2.

    (ii) According to the model, what proportion of customers take more than four

    minutes to serve?

    Find the probability that it takes between five and ten minutes to serve a

    customer.

    (iii) Use the p.d.f. f(t) to calculate the mean and variance of the time taken to serve a customer.

    (iv) Use Formula (4.1) of Unit 1 to calculate the mean time taken to serve a

    custome.

    Formula (4.1) = (see below)

    (v) Simulate the time taken to serve a customer, using the random number

    u=0.5536, which is an observation from the uniform distribution U(0,1).

    Give your answer in minutes and seconds, to the nearest second.

    foruma attached.
    (i) F(t) = \int_{2}^{t} 64 u^{-5} \, du.

    (ii) \Pr(T > 4) = \int_{4}^{+\infty} 64 t^{-5} \, dt. Alternatively, calculate 1 - \Pr(T < 4) = 1 - F(4).

    \Pr(5 < T < 10) = \int_{5}^{10} 64 t^{-5} \, dt. Alternatively, calculate \Pr(T < 10) - \Pr(T < 5) = F(10) - F(5).

    (iii) E(T) = \int_{2}^{+\infty} t (64 t^{-5}) \, dt = \int_{2}^{+\infty} 64 t^{-4} \, dt  .

    Var(T) = E(T^2) - [E(T)]^2 where E(T^2) = \int_{2}^{+\infty} t^2 (64 t^{-5}) \, dt = \int_{2}^{+\infty} 64 t^{-3} \, dt.

    (iv) Calculate \int_2^{+\infty} 1 - \left[ 1 - 16t^{-4}\right] \, dt.

    (v) Read http://www.mathhelpforum.com/math-he...verse-cdf.html and http://www.mathhelpforum.com/math-he...functions.html.
    Last edited by mr fantastic; March 8th 2009 at 05:09 AM.
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  3. #3
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    Thanks so much MR. F it has been the great help already.

    here is the question on print screen.
    Attached Thumbnails Attached Thumbnails Homework - Probability Density Function-question-4.jpg  
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  4. #4
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    Quote Originally Posted by wallace View Post
    At my local sandwich bar, I have noticed that it always takes at least two minutes to serve a customer and it can take much longer to fulfil an unusual order. The time in minutes taken to serve a customer may be modelled by a continuous random variable T with probability density function f(t)= 64 t5 ,t≥ 2.
    (i) Show that the c.d.f. of the random variable T is given by
    F(t)= 1 − 16 t4 ,t≥ 2.

    (ii) According to the model, what proportion of customers take more than four
    minutes to serve?
    Find the probability that it takes between five and ten minutes to serve a
    customer.

    (iii) Use the p.d.f. f(t) to calculate the mean and variance of the time taken to serve a customer.

    (iv) Use Formula (4.1) of Unit 1 to calculate the mean time taken to serve a
    custome.
    Formula (4.1) = (see below)
    (v) Simulate the time taken to serve a customer, using the random number
    u=0.5536, which is an observation from the uniform distribution U(0,1).
    Give your answer in minutes and seconds, to the nearest second.

    foruma attached.
    These questions form part of a graded assessment. Thread closed.
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