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In the carnival game Under-or -Over-Seven, a pair of fair dice is rolled once, and the resulting sum determines whether the player wins or loses his or her bet. For example, the player can bet
a. Construct the
b. Construct the probability distribution representing the different outcome that are possible for
c. Construct the probability distribution representing the different outcome that are possible for
d. show that the expected long-run profit (or loss) to the player is the same, no matter which method to play is used.
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Chapter 5 Solutions
Basic Business Statistics
- 3. Let X1, X2,..., X, be independent, Exp(1)-distributed random variables, and set V₁₁ = max Xk and W₁ = X₁+x+x+ Isk≤narrow_forward7. Consider the function (t)=(1+|t|)e, ER. (a) Prove that is a characteristic function. (b) Prove that the corresponding distribution is absolutely continuous. (c) Prove, departing from itself, that the distribution has finite mean and variance. (d) Prove, without computation, that the mean equals 0. (e) Compute the density.arrow_forward1. Show, by using characteristic, or moment generating functions, that if fx(x) = ½ex, -∞0 < x < ∞, then XY₁ - Y2, where Y₁ and Y2 are independent, exponentially distributed random variables.arrow_forward
- 1. Show, by using characteristic, or moment generating functions, that if 1 fx(x): x) = ½exarrow_forward1990) 02-02 50% mesob berceus +7 What's the probability of getting more than 1 head on 10 flips of a fair coin?arrow_forward9. The concentration function of a random variable X is defined as Qx(h) sup P(x≤x≤x+h), h>0. = x (a) Show that Qx+b(h) = Qx(h).arrow_forward
- Suppose that you buy a lottery ticket, and you have to pick six numbers from 1 through 50 (repetitions allowed). Which combination is more likely to win: 13, 48, 17, 22, 6, 39 or 1, 2, 3, 4, 5, 6? barrow_forward2 Make a histogram from this data set of test scores: 72, 79, 81, 80, 63, 62, 89, 99, 50, 78, 87, 97, 55, 69, 97, 87, 88, 99, 76, 78, 65, 77, 88, 90, and 81. Would a pie chart be appropriate for this data? ganizing Quantitative Data: Charts and Graphs 45arrow_forward10 Meteorologists use computer models to predict when and where a hurricane will hit shore. Suppose they predict that hurricane Stat has a 20 percent chance of hitting the East Coast. a. On what info are the meteorologists basing this prediction? b. Why is this prediction harder to make than your chance of getting a head on your next coin toss? U anoiaarrow_forward
- 6. Show that one cannot find independent, identically distributed random variables X and Y such that X-YE U(-1, 1). 7 Consider al onarrow_forward2 Which situation is more likely to produce exactly 50 percent heads: flipping a coin 10 times or flipping a coin 10,000 times?arrow_forward2. Let X1, X2..... X, be independent random variables with expectation 0 and finite third moments. Show, with the aid of characteristic functions, that E(X₁ + X₂++X)³ = EX² + EX²++ EX₁₂-arrow_forward
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