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Math 425 Fall 2024 - HW 13

Due Friday 11/29, 11:59pm, via Gradescope

Please note:

(1). Please include detailed steps. Only providing the result will not get full credits.

(2). Please write at most one problem in each page. If you reach the bottom please start a new page instead of writing two columns in one page. If a problem contains multiple small questions, you may write them in one page.

(3). Please associate pages with problems in Gradescope.

1. Suppose X = Uniform(−1, 2). When X = x, suppose Y is uniformly dis-trubuted between x and 2x. Use conditioning to find E[Y].

Notice: Be careful with the integration bounds when x < 0.

2. Choose an integer randomly among {1, 2, 3, 4, 5}, denoted by random variable X. Then choose an integer randomly among {k : 1 ≤ k ≤ X}, denoted by Y . Use conditioning to find E[Y].

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

Hint: Let Xi be the result of the i-th rolling. We need to approximate P(X1 + · · · + X79 ≤ 300) using the central limit theorem.

4. Amy has 100 light bulbs whose lifetimes are independent and identical ex-ponential random variables with an expectation of 5 hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one (ignore the time for replacing), approximate the probability that there is still a working bulb after 525 hours.

Hint: Let Xi be the working time of the i-th light bulb. We need to approximate P(X1 + · · · + X100 ≥ 525).

5. Engineers believe that W, the amount of weight (in units of 1000 pounds) that a bridge can withstand without structural damage resulting, is normally distributed with mean 400 and variance 402. Suppose that the weight (in units of 1000 pounds) of a car is a random variable with mean 3 and variance 0.32. Approximately how many cars would have to be on the bridge for the probability of structural damage to exceed 0.1?

6. An insurance company has 10,000 automobile policyholders. The expected yearly claim per policyholder is 240 dollars, with a standard deviation of 800. Approximate the probability that the total yearly claim exceeds 2.7 million dollars.

7. We have 100 components that we will put in use in a sequential fashion. That is, component 1 is initially put in use, and upon failure, it is replaced by component 2, which is itself replaced upon failure by component 3, and so on. If the lifetime (in hours) of component i is exponential random variable with mean 10 + 10/i. Use Markov’s inequality to estimate the probability that the total life of all components will exceed 1200 hours.

8. If X has mean µ and standard deviation σ, the ratio r = σ/|µ| is called the measurement signal-to-noise ratio of X. The idea is that X can be expressed as X = µ + (X − µ), with µ representing the signal and X − µ the noise. If we

define D = |µ/X−µ| as the relative deviation of X from its signal (or mean) µ, show that for α > 0 we have

P{D ≤ α} ≥ 1 − r2α2/1