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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
ISBN: 9781305251809

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BuyFindarrow_forward

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
ISBN: 9781305251809
Textbook Problem

There are two traffic lights on a commuter’s route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose these two variables are independent, each with pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2).

x1 0 1 2
p(x1) 0.2 0.5 0.3

    µ =1.1, σ2=0.49

  1. a. Determine the pmf of To = X1 + X2.
  2. b. Calculate µTo. How does it relate to µ, the population mean?
  3. c. Calculate σ2 To. How does it relate to σ2, the population variance?
  4. d. Let X3 and X4 be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With To = the sum of all four Xi’s, what now are the values of E(To) and V(To)?
  5. e. Referring back to (d), what are the values of P(T0 = 8) and P(To ≥ 7) [Hint: Don’t even think of listing all possible outcomes!]

a.

To determine

Find the pmf of T0=X1+X2.

Explanation

Given info:

The random variables X1 and X2 denote respectively the number of lights a commuter has to stop at to get to work and that while returning from work. Both independently take values 0, 1, 2 with respective probabilities 0.2, 0.5, 0.3, with μ=1.1, σ2=0.49.

Calculation:

Independence of random variables:

Two discrete random variables, X and Y, are said to be independent, if their joint pmf, p(x,y) can be expressed as the product of the marginal pmf of the individual random variables, pX(x) and pY(y), that is, if:

p(x,y)=pX(x)pY(y).

The distributions of X1 and X2 are independent.

Thus, for each pair of value (x1,x2) taken by X1 and X2,

p(x1,x2)=pX1(x1)pX2(x2).

For the pair of values (0, 0), the joint pmf is:

p(0,0)=pX1(0)pX2(0)=0.2×0.2=0.04

b.

To determine

Calculate μT0 and give its relation with the population mean, μ.

c.

To determine

Calculate σ2T0 and give its relation with the population variance, σ2.

d.

To determine

Find the new values of E(T0) and V(T0).

e.

To determine

Find the values of P(T0=8) and P(T07).

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