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Module 7 (Chapter 6) Assignment Problem 1 The random variable x is known to be uniformly distributed between 1.0 and 1.5. a. Show the graph of the probability density function. To create a probability density graph, you first determine the height of the graph between 1.0 and 1.5. The formula to find the height of the graph is f(x) = 1 / (b - a), where B is the highest value (1.5) and A is the lowest value (1.0). Once you find the height, the uniform Probability Distribution shape is always a triangle. So, you plot the points for 1 and 1.5 at 2 and draw a rectangle. b. Compute 𝑃(𝑥 = 1.25) . P(x = 1.25) = 0. This is because x is continuously distributed in a uniform probability distribution. c. Compute 𝑃(1.0 ≤ 𝑥 ≤ 1.25) . The probability for (1.0 ≤ 𝑥 ≤ 1.25) is 0.5. To find this probability, you first determine the x value for P(X ≤ 1.25), which is 2.5. Then, you calculate the values for P(1.0 ≤ 𝑥 ), which is 2. Subtracting P(X ≤ 1.25) - P(1.0 ≤ 𝑥 ) yields the value of 0.5. d. Compute 𝑃(1.20 < 𝑥 < 1.5) . P(1.20 < 𝑥 < 1.5) is 0.6. To find this probability, you first determine the x value for P(x < 1.5), which is 3.0. Then you find the x value for P(1.20 < x), which is 2.4. Subtracting P(x < 1.5) - P(1.20 < x) yields the value of 0.6. Problem 2 Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that the highest bid in excess of $10,000 will be accepted. Assume that the competitor’s bid x is a random variable that is uniformly distributed between $10,000 and $15,000. a. Suppose you bid $12,000. What is the probability that your bid will be accepted? The probability that my bid will be accepted if I bid $12,000 is 0.04. To find this probability, you can use the formula (b - a) / (b - a), where the only variable that changes is "b." In the numerator, you subtract the "b" value from the lowest value, which is $10,000. So, in this case, it would be 12,000 - 10,000.In the denominator, you use the "b" value listed as the highest value in the uniform probability distribution, which is $15,000. Using this formula will give you the probability of your bid being accepted. 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 0 0.5 1 1.5 2 2.5 PROBABILITY DENSITY FUNCTION
b. Suppose you bid $14,000. What is the probability that your bid will be accepted? The probability that my bid will be accepted if I bid $14,000 is 0.08. To find this probability, you can use the formula (b - a) / (b - a), where the only variable that changes is "b." In the numerator, you subtract the "b" value from the lowest value, which is $10,000. So, in this case, it would be 14,000 - 10,000.In the denominator, you use the "b" value listed as the highest value in the uniform probability distribution, which is $15,000. Using this formula will give you the probability of your bid being accepted. c. What amount should you bid to maximize the probability that you get the property? To maximize the probability of acquiring the property, you should place the highest bid within the Uniform Probability Distribution. To maximize your probability of success, aim to get as close as possible to a value of 1. Therefore, the recommended bid amount would be $15,000. This will yield a probability of 1, which can be calculated by dividing (15,000 - 10,000) by (15,000 - 10,000) resulting in 1. d. Suppose you know someone who is willing to pay you $16,000 for the property. Would you consider bidding less than the amount in part (c)? Why or why not? I would refrain from placing a bid lower than the amount specified in part c. Bidding a smaller amount would lead to a decrease in the expected profit, as it would lower the probability of successfully buying the property. By offering $15,000, I would secure the highest possible probability of winning and satisfying the bidders requirements. Problem 3 Given that z is a standard normal random variable, compute the following probabilities. a. 𝑃(𝑧 ≤ −1.0) The probability P(z ≤ −1.0) is 0.15865525. There are two ways to find this probability: using a z-scores table or using the formula in Excel, which is the normal distribution function. To calculate the z-score, you can use the formula z = (x - µ) / σ. b. 𝑃(𝑧 ≥ −1) The probability of (z ≥ −1) is 0.84134475. To find this probability, you can subtract the probability of P( 𝑧 ≤ −1.0) from 1. c. 𝑃(𝑧 ≥ −1.5) The probability of (z ≥ −1.5) is 0.9331928. To find this probability, you can subtract the probability of P( 𝑧 ≤ −1.5) from 1. d. 𝑃(−2.5 ≤ 𝑧) The probability of (z ≥ −2.5) is 0.99379033. To find this probability, you can subtract the probability of P( 𝑧 ≤ −2.5) from 1. e. 𝑃(−3 < 𝑧 ≤ 0) The probability of 𝑃 (−3 < z ≤ 0) is 0.4986501. To find this probability, subtract the probability of P( 𝑧 ≤ 0) from P(−3 < 𝑧 ). Problem 4 In an article about the cost of health care, Money magazine reported that a visit to a hospital emergency room for something as simple as a sore throat has a mean cost of $328 ( Money , January 2009). Assume that the cost for this type of hospital emergency room visit is normally distributed with a standard deviation of $92. Answer the following questions about the cost of a hospital emergency room visit for this medical service.
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