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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 4: Q.4.37 (page 166)

Find Var(X) and Var(Y) for X and Y as given in Problem 4.21

Short Answer

Expert verified

In the given information the answers are $V a r (X) = 80.92, V a r (Y) = 84.5$

Step by step solution

01

Step 1:Given Information

$V a r (x) = E (X^{2}) - E {(X)}^{2}$

02

Step 2:Calculation

By using the result d\from problem 22 and same notation, we can compute the variance using the identity $Var (X) = E (X^{2}) - E (X)^{2}$ .

Now we have that

E (X^{2}) = 40^{2} \cdot \frac{40}{148} + 33^{2} \cdot \frac{33}{148} + 25^{2} \cdot \frac{25}{148} + 50^{2} \cdot \frac{50}{148} = 1625.42

Then we have that,

Var (X) = E (X^{2}) - E (X)^{2} = 1625.42 - 39 . 3^{2} = 80.92

Similarly for $Y$ ,

E (Y^{2}) = 40^{2} \cdot \frac{1}{4} + 33^{2} \cdot \frac{1}{4} + 25^{2} \cdot \frac{1}{4} + 50^{2} \cdot \frac{1}{4} = 1453.5

Now, we have that

Var (Y) = E (Y^{2}) - E (Y)^{2} = 1453.5 - 37^{2} = 84.5

03

Step 3:Final Answer

The value of $Var (X) = 80.92, Var (Y) = 84.5$

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Most popular questions from this chapter

Let $X$ be a Poisson random variable with parameter $λ$ . Show that $P X = i$ increases monotonically and then decreases monotonically as $i$ increases, reaching its maximum when $i$ is the largest integer not exceeding $λ$ .

Hint: Consider $P X = i / P X = i - 1$ .

A coin that when flipped comes up heads with probability $p$ is flipped until either heads or tails has occurred twice. Find the expected number of flips

Consider n independent trials, each of which results in one of the outcomes $1, . . ., k$ with respective probabilities $p_{1}, \dots, p_{k}, \sum_{i = 1}^{k} p_{i} = 1$ Show that if all the $p i$ are small, then the probability that no trial outcome occurs more than once is approximately equal to $\exp (- n (n - 1) \sum_{i} p_{i}^{2} / 2)$ .

Here is another way to obtain a set of recursive equations for determining $P_{n}$ , the probability that there is a string of $k$ consecutive heads in a sequence of $n$ flips of a fair coin that comes up heads with probability $p$ :

(a) Argue that for $k < n$ , there will be a string of $k$ consecutive heads if either

1. there is a string of $k$ consecutive heads within the first $n - 1$ flips, or

2. there is no string of $k$ consecutive heads within the first $n - k - 1$ flips, flip $n - k$ is a tail, and flips $n - k + 1, \dots, n$ are all heads.

(b) Using the preceding, relate $P_{n} t o P_{n - 1}$ . Starting with $P_{k} = p^{k}$ , the recursion can be used to obtain $P_{k + 1}$ , then $P_{k + 2}$ , and so on, up to $P_{n}$ .

People enter a gambling casino at a rate of $1$ every $2$ minutes.

$(a)$ What is the probability that no one enters between $12 : 00$ and $12 : 05$ ?

$(b)$ What is the probability that at least $4$ people enter the casino during that time?

See all solutions

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