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Chapter 8: Q 8.6 (page 390)

A die is continually rolled until the total sum of all rolls exceeds 300. Approximate the probability that at least 80 rolls are necessary.

Short Answer

Expert verified

P{Y79鈮�300}=0.9429

Step by step solution

01

Step 1. Given information

Let Xidenote the value of the die, of ithroll.

02

Step 2. Mean and variance of Xi

Mean = E(Xi)=(1+2+3+4+5+6)6=216=72

Variance =role="math" localid="1648052296820" V(Xi)=3512

03

Step 3. Introducing Yn

Assuming that die is continually rolled until the total sum of all rolls exceeds 300.

Let Ynrepresents the total sum of all rolls with n number of rolls.

Yn=鈭�Xi1n

04

Step 4. Mean and variance of Yn

Mean = E(Yn)=n脳E(Xi)=7n2

Variance =V(Yn)=n脳V(Xi)=35n12

05

Step 5. To find

The probability that at least 80 rolls are necessary can be written as -

P鈭�Xi鈮�300179=PY79鈮�300

06

Step 6. Using central limit theorem

P{Y79鈮�300}=PY79-7(79)235(79)12鈮�300-7(79)235(79)12P{Y79鈮�300}=PZ鈮�1.58P{Y79鈮�300}=0.9429

07

Step 7. Final answer

The probability that at least 80 rolls are necessary is 0.9429

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Most popular questions from this chapter

ItXis a Poisson random variable with a mean100, thenPX>120is approximately

(a).02,

(b).5or

(c).3?

Let X1, ... , X20 be independent Poisson random variables with mean 1.

(a) Use the Markov inequality to obtain a bound on

P鈭�Xi>15120

(b) Use the central limit theorem to approximate

P鈭�Xi>15120

Fifty numbers are rounded off to the nearest integer and then summed. If the individual round-off errors are uniformly distributed over (鈭�.5, .5), approximate the probability that the resultant sum differs from the exact sum by more than 3.

It Xis a Poisson random variable with a mean位, showing that fori<位,

P{X鈮�i}鈮�e-位(e位)iii

Let f(x)be a continuous function defined for0鈮�x鈮�1. Consider the functions

Bn(x)=鈭�k=0nfknnkxk(1鈭�x)n-k(called Bernstein polynomials) and prove that

limn鈫�鈭�Bn(x)=f(x).

Hint: Let X1,X2,...be independent Bernoulli random variables with meanx. Show that

Bn(x)=Efx1+...+xnn

and then use Theoretical Exercise8.4.

Since it can be shown that the convergence of Bn(x)to f(x)is uniformx, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.
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