You Decide To Move Out Of Your College Dorms And Get An Apa

You Decide To Move Out Of Your Colleges Dorms And Get An Apartment A

You decide to move out of your college's dorms and get an apartment, and you want to discuss the budget with your roommate. You know that your monthly grocery bill G will depend on several factors, and G is approximately normally distributed with a mean given by the formula: μ = 300 + 10M - 100B + 50H, where M is the number of meals to which you invite guests, B is a measure of how busy you are (assumed to be uniformly distributed over [0,1]), and H is an indicator variable that is 1 during holiday months (November, December, January) and 0 otherwise. Your task is to determine the expected value E[G] for a randomly chosen month based on this information.

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The problem involves calculating the expected monthly grocery bill, G, based on its approximate normal distribution and understanding how the mean of this distribution varies with certain factors. The primary task is to find the expected value of G, E[G], taking into account the known relationship between G's mean and the variables M, B, and H.

Given that G follows a normal distribution with variance 2500 and mean μ = 300 + 10M - 100B + 50H, the expectation of G, E[G], is equal to the expectation of its mean, E[μ], because the properties of expectation allow us to write E[G] = E[μ] when the variance is finite and μ is a random variable depending on M, B, and H.

The mean μ depends on three variables: M, B, and H. Each variable's expectation contributes to the overall expectation of G. Assuming linearity of expectation, we have:

E[G] = E[μ] = E[300 + 10M - 100B + 50H] = 300 + 10E[M] - 100E[B] + 50E[H].

Let's examine each component:

1. E[M]: M is the number of invited guests' meals, with an expected value E[M] = 8 (given in the problem).

2. E[B]: B is uniformly distributed over [0, 1], so its expected value is:

E[B] = (a + b)/2 = (0 + 1)/2 = 0.5.

3. E[H]: H is an indicator variable, which equals 1 during holiday months and 0 otherwise. Assuming a uniform probability for observing any month (or given exact months), the probability that a randomly chosen month is a holiday month (November, December, January) is 3 out of 12 months, i.e., 1/4. Therefore:

E[H] = P(H=1) = 1/4 = 0.25.

Putting these together:

E[G] = 300 + 10 8 - 100 0.5 + 50 * 0.25 = 300 + 80 - 50 + 12.5 = 342.5.

Among the options provided, the expression that matches this calculation is:

c) 300 + 10E[M] - 100E[B] + 50E[H], which equals 342.5 after substituting the expected values.

In summary, the expected grocery bill for a randomly chosen month can be calculated using the linearity of expectation and the known distributions of the contributing variables.

References

  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Mandelbrot, B. (1992). The Fractal Geometry of Nature. W. H. Freeman.
  • Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.

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