Statistics and Their Distributions

Transcription

1 Statistics and Their Distributions Deriving Sampling Distributions Example A certain system consists of two identical components. The life time of each component is supposed to have an expentional distribution with parameter λ. The lifetime of the system is supposed to be the sum of the two components and the two components are assumed to work independently. Let X 1 and X 2 be the lifetime of the two components, respectively. What can we say about the lifetime of the system T 0 = X 1 + X 2?

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3 Proposition Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 1.Whether or not the X i s are independent, E(a 1 X 1 + a 2 X a n X n ) = a 1 E(X 1 ) + a 2 E(X 2 ) + + a n E(X n ) 2. If X 1, X 2,..., X n are independent, = a 1 µ 1 + a 2 µ a n µ n V (a 1 X 1 + a 2 X a n X n ) = a 2 1V (X 1 ) + a 2 2V (X 2 ) + + a 2 nv (X n ) = a 1 σ a 2 σ a n σ 2 n

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5 Proposition (Continued) Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 3. More generally, for any X 1, X 2,..., X n V (a 1 X 1 + a 2 X a n X n ) = n n a i a j Cov(X i, X j ) i=1 j=1

6 Proposition (Continued) Let X 1, X 2,..., X n have mean values µ 1, µ 2,..., µ n, respectively, and variances σ 2 1, σ2 2,..., σ2 n, respectively. 3. More generally, for any X 1, X 2,..., X n V (a 1 X 1 + a 2 X a n X n ) = n n a i a j Cov(X i, X j ) i=1 j=1 We call a 1 X 1 + a 2 X a n X n a linear combination of the X i s.

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8 Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time.

9 Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time?

10 Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time?

11 Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time? c. What are the expected value and variance of the difference between morning and evening waiting times on a given day?

12 Example (Problem 64) Suppose your waiting time for a bus in the morning is uniformly distributed on [0,8], whereas waiting time in the evening is uniformly distributed on [0,10] independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time? b. What is the variance of your total waiting time? c. What are the expected value and variance of the difference between morning and evening waiting times on a given day? d. What are the expected value and variance of the difference between total morning waiting time and total evening waiting time on a particular week?

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14 Corollary E(X 1 X 2 ) = E(X 1 ) E(X 2 ) and, if X 1 and X 2 are independent, V (X 1 X 2 ) = V (X 1 ) + V (X 2 ).

15 Corollary E(X 1 X 2 ) = E(X 1 ) E(X 2 ) and, if X 1 and X 2 are independent, V (X 1 X 2 ) = V (X 1 ) + V (X 2 ). Proposition If X 1, X 2,..., X n are independent, normally distributed rv s (with possibly different means and/or variances), then any linear combination of the X i s also has a normal distribution. In particular, the difference X 1 X 2 between two independent, normally distributed variables is itself normally distributed.

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17 Example (Problem 62) Manufacture of a certain component requires three different maching operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 30, and 20min, respectively, and the standard deviations are 1, 2, and 1.5min, respectively.

18 Example (Problem 62) Manufacture of a certain component requires three different maching operations. Machining time for each operation has a normal distribution, and the three times are independent of one another. The mean values are 15, 30, and 20min, respectively, and the standard deviations are 1, 2, and 1.5min, respectively. What is the probability that it takes at most 1 hour of machining time to produce a randomly selected component?