Monthly Archives: April 2015

Practice problems for order statistics and multinomial probabilities

This post presents exercises on calculating order statistics using multinomial probabilities. These exercises are to reinforce the calculation demonstrated in this blog post.

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Practice Problems

Practice Problems 1
Draw a random sample X_1,X_2,\cdots,X_{11} of size 11 from the uniform distribution U(0,4). Calculate the following:

  • P(Y_4<2<Y_5<Y_7<4<Y_8)
  • P(Y_4<2<Y_6<Y_7<4<Y_8)

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Practice Problems 2
Draw a random sample X_1,X_2,\cdots,X_7 of size 7 from the uniform distribution U(0,5). Calculate the probability P(Y_4<2<4<Y_7).

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Practice Problems 3
Same setting as in Practice Problem 2. Calculate P(Y_7>4 \ | \ Y_4<2) and P(Y_7>4). Compare the conditional probability with the unconditional probability. Does the answer for P(Y_7>4 \ | \ Y_4<2) make sense in relation to P(Y_7>4)?

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Practice Problems 4
Same setting as in Practice Problem 2. Calculate the following:

  • P(Y_4<2<Y_7<4)
  • P(2<Y_7<4 \ | \ Y_4<2)
  • P(2<Y_7<4)
  • Does the answer for P(2<Y_7<4 \ | \ Y_4<2) make sense in relation to P(2<Y_7<4)?

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Practice Problems 5
Draw a random sample X_1,X_2,\cdots,X_6 of size 6 from the uniform distribution U(0,4). Consider the conditional distribution Y_3 \ | \ Y_5<2. Calculate the following:

  • P(Y_3 \le t \ | \ Y_5<2)
  • f_{Y_3}(t \ | \ Y_5<2)
  • E(Y_3 \ | \ Y_5<2)
  • E(Y_3)

where 0<t<2. Compare E(Y_3) and E(Y_3 \ | \ Y_5<2). Does the answer for the conditional mean make sense?

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Practice Problems 6
Draw a random sample X_1,X_2,\cdots,X_7 of size 7 from the uniform distribution U(0,5). Calculate the following:

  • P(Y_4 > 4 \ | \ Y_2>2)
  • P(Y_4 > 4)
  • Compare the two probabilities. Does the answer for the conditional probability make sense?

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Answers

Practice Problems 1

  • \displaystyle \frac{11550}{177147}
  • \displaystyle \frac{18480}{177147}

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Practice Problems 2

  • \displaystyle \frac{11088}{78125}

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Practice Problems 3

  • \displaystyle P(Y_7>4 \ | \ Y_4<2)=\frac{11088}{22640}
  • \displaystyle P(Y_7>4)=\frac{61741}{78125}

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Practice Problems 4

  • \displaystyle P(Y_4<2<Y_7<4)=\frac{8064}{78125}
  • \displaystyle P(2<Y_7<4 \ | \ Y_4<2)=\frac{8064}{22640}
  • \displaystyle P(2<Y_7<4)=\frac{16256}{78125}

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Practice Problems 5

  • \displaystyle P(Y_3 \le t \ | \ Y_5<2)=\frac{-10t^6+84t^5-300t^4+400t^3}{448}
  • \displaystyle f_{Y_3}(t \ | \ Y_5<2)=\frac{-60t^5+420t^4-1200t^3+1200t^2}{448}
  • \displaystyle E(Y_3 \ | \ Y_5<2)=\frac{55}{49}
  • \displaystyle E(Y_3)=\frac{84}{49}

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Practice Problems 6

  • \displaystyle \frac{3641}{12393}
  • \displaystyle \frac{2605}{78125}

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\copyright \ 2015 \text{ by Dan Ma}

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Calculating the probability distributions of order statistics

This post presents exercises on finding the probability distributions of order statistics to complement a discussion of the same topic.

Consider a random sample X_1,X_2,\cdots,X_n drawn from a continuous distribution with common distribution function F(x). The order statistics Y_1<Y_2<\cdots <Y_n are obtained by ranking the sample items in increasing order. In this post, we present some exercises to complement this previous post. The thought processes illustrated by these exercises will be helpful in non-parametric inference, specifically in the construction of confidence intervals for unknown population percentiles.

In the problems that follow, Y_1<Y_2<\cdots <Y_n are the order statistics that arise from the random sample X_1,X_2,\cdots,X_n. There are two ways to work with the probability distribution of an order statistic Y_j. One is to find the distribution function F_{Y_j}(y)=P(Y_j \le y). Once this is obtained, the density function f_{Y_j}(y) is derived by taking derivative. Another way is to derive f_{Y_j}(y) directly.

We assume that the random sample X_1,X_2,\cdots,X_n is drawn from a probability distribution with distribution function F(x)=P(X \le x) and with density function f(x). To compute F_{Y_j}(y)=P(Y_j \le y), note that for the event Y_j \le y to occur, at least j many sample items X_i are less than y. So the random drawing of each sample item is a Bernoulli trial with probability of success F(y)=P(X \le y). Thus F_{Y_j}(y)=P(Y_j \le y) is the following probability computed from a binomial distribution.

    \displaystyle F_{Y_j}(y)=P(Y_j \le y)=\sum \limits_{k=j}^n \ \binom{n}{k} \ F(y)^k \ \biggl[1-F(y) \biggr]^{n-k} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

Once the distribution function F_{Y_j}(y) is found, the density function f_{Y_j}(y) can be derived by taking derivative on F_{Y_j}(y). The density function f_{Y_j}(y) can also be obtained directly by this thought process. Think of the density function f_{Y_j}(y) as the probability that the jth order statistic Y_j is right around y. So there must be j-1 sample items less than y and n-j sample items above y. One way this can happen is:

    \displaystyle F(y)^{j-1} \ f(y) \ \biggl[1-F(y) \biggr]^{n-j}

The first term is the probability that j-1 sample terms are less than y. The second term is the probability that one sample item is right around y. The third term is the probability that n-j sample items are above y. But this is only one way. To capture all possibilities, we multiply it by the multinomial coefficient.

    \displaystyle f(_{Y_j}(y)=\frac{n!}{(j-1)! \ 1! \ (n-j)!} \ F(y)^{j-1} \ f(y) \ \biggl[1-F(y) \biggr]^{n-j} \ \ \ \ \ \ \ \ \ \ (2)

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Practice Problems

Practice Problems 1
Draw a random sample X_1,X_2,\cdots,X_8 of size 8 from the uniform distribution U(0,4). Calculate the probability P(Y_3>3) where Y_3 is the third order statistic.

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Practice Problems 2
Draw a random sample X_1,X_2,X_3,X_4,X_5 of size 5 from a continuous distribution with density function f(x)=\frac{x}{2} where 0<x<2. Find the probability that the sample median is less than 1.

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Practice Problems 3
Draw a random sample X_1,X_2,X_3,X_4,X_5 of size 5 from the uniform distribution U(0,4).

  1. Calculate the distribution function P(Y_4 \le y) for the fourth order statistic. Then differentiate it to obtain the density function f_{Y_4}(y) of Y_4.
  2. Use the thought process behind formula (2) above to directly write down the density function f_{Y_4}(y) directly.

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Practice Problems 4
Draw a random sample X_1,X_2,\cdots,X_9 of size 9 from an exponential distribution with mean \frac{1}{\alpha}.

  1. Calculate the distribution function P(Y_1 \le y) for the first order statistic (the minimum). Then differentiate it to obtain the density function f_{Y_1}(y) of Y_1.
  2. Use the thought process behind formula (2) above to directly write down the density function f_{Y_1}(y) directly.

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Practice Problems 5
Draw a random sample X_1,X_2,\cdots,X_9 of size 9 from an exponential distribution with mean \frac{1}{\alpha}.

  1. Calculate the distribution function P(Y_2 \le y) for the second order statistic. Then differentiate it to obtain the density function f_{Y_2}(y) of Y_2.
  2. Use the thought process behind formula (2) above to directly write down the density function f_{Y_2}(y) directly.

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Practice Problems 6
Draw a random sample X_1,X_2,\cdots,X_8 of size 8 from the uniform distribution U(0,1). Find E(Y_6), the expected value of the sixth order statistic.

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Practice Problems 7
Draw a random sample X_1,X_2,\cdots,X_{10} of size 10 from the uniform distribution U(0,1). Find Var(Y_9), the variance of the ninth order statistic.

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Practice Problems 8
Draw a random sample X_1,X_2,\cdots,X_{6} of size 6 from a population whose 25th percentile is 83. Find the probability the third order statistic Y_3 is less than 83.

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Practice Problems 9
Draw a random sample X_1,X_2,\cdots,X_{6} of size 6 from a population whose 75th percentile is 105. Find the probability the third order statistic Y_3 is less than 105.

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Practice Problems 10
Draw a random sample X_1,X_2,X_3,X_4 of size 4 from an exponential distribution with mean 10. Calculate P(Y_4>15 \ | \ Y_4 > 10).

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Practice Problems 11
Draw a random sample X_1,X_2,X_3,X_4,X_5 of size 5 from a continuous distribution with density function f(x)=\frac{x}{2} where 0<x<2. Find E(Y_5), the expected value of the sample maximum.

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Practice Problems 12
Draw a random sample of size 12 from an exponential distribution with mean 2. Calculate P(Y_1>0.5 | Y_1 > 0.25).

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Answers

Practice Problems 1

  • \displaystyle P(Y_3>3)=\frac{277}{65536}

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Practice Problems 2

  • \displaystyle P(Y_3<1)=\frac{106}{1024}

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Practice Problems 3

  • \displaystyle f_{Y_4}(y)=20 \ \biggl( \frac{y}{4} \biggr)^3 \ \frac{1}{4} \ \biggl(1- \frac{y}{4} \biggr)

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Practice Problems 4

  • \displaystyle f_{Y_1}(y)=9 \alpha \ e^{-9 \alpha y}

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Practice Problems 5

  • \displaystyle f_{Y_2}(y)=72 \ \biggl(1-e^{-\alpha y} \biggr) \ \alpha e^{-\alpha y} \ \biggl( e^{-\alpha y} \biggr)^7

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Practice Problems 6

  • \displaystyle E(Y_6)=\frac{2}{3}

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Practice Problems 7

  • \displaystyle Var(Y_9)=\frac{3}{242}

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Practice Problems 8

  • \displaystyle P(Y_3<83)=\frac{694}{4096}

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Practice Problems 9

  • \displaystyle P(Y_3<105)=\frac{3942}{4096}

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Practice Problems 10

  • \displaystyle P(Y_4>15 \ | \ Y_4 > 10) = 0.756546693

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Practice Problems 11

  • \displaystyle E(Y_5)=\frac{20}{11}

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Practice Problems 12

  • \displaystyle P(Y_1>0.5 | Y_1 > 0.25)=e^{-1.5}

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\copyright \ 2015 \text{ by Dan Ma}