Suppose your waiting time for a bus in the morning is...

Suppose your waiting time for a bus in the morning is uniformly distributed on $[0,8],$ whereas waiting time in the evening is uniformly distributed on $[0,10]$ independent of morning waiting time. (a) If you take the bus each morning and evening for a week, what is your total expected waiting time? [Hint: Define rvs $X_{1}, \ldots, X_{10}$ and use a rule of expected value.] (b) What is the variance of your total waiting time? (c) What are the expected value and variance of the difference between morning and evening waiting times on a given day? (d) What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?

Suppose your waiting time for a bus in the morning is uniformly distributed on $[0,8],$ whereas
waiting time in the evening is uniformly distributed on $[0,10]$ independent of morning waiting
time.
(a) If you take the bus each morning and evening for a week, what is your total expected waiting time? [Hint: Define rvs $X_{1}, \ldots, X_{10}$ and use a rule of expected value.]
(b) What is the variance of your total waiting time?
(c) What are the expected value and variance of the difference between morning and evening waiting times on a given day?
(d) What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week?

Key Concepts

Uniform Distribution

The uniform distribution is a continuous probability distribution where every outcome in a given interval is equally likely. This concept is important here as it provides straightforward formulas for calculating the expectation (the midpoint of the interval) and the variance (which measures the spread of the distribution) for the waiting times.

Linearity of Expectation

Linearity of expectation is a fundamental property stating that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent. This concept simplifies the calculation of total expected waiting times over several independent events.

Variance of Independent Random Variables

When dealing with independent random variables, the variance of their sum is the sum of their individual variances. This property is crucial for calculating the total variance of waiting times across different time periods or days as it allows one to add the variances directly.

Transformation of Random Variables (Differences)

When subtracting one random variable from another, the expected value of the difference is the difference of the expected values, while the variance of the difference, under independence, is the sum of the variances. This concept is used to analyze the differences between waiting times in the morning and evening.

Transcript

00:02 We're told that waiting time for a bus in the morning is uniformly distributed on the interval 08, and waiting time in the evening is uniformly distributed on the interval 010, independent of the morning waiting time.

00:21 In part a, suppose that we take the bus each morning and evening for a week, we're asked to find our total expected waiting time.

00:39 So, first let's define some random variables.

00:44 So let x1 through x5, these five random variables be the morning times during this week, and x6 through x10 be the evening times.

01:14 We have that we want to find the total expected waiting time.

01:22 This is the expected value of the sum of x1 through x1.

01:26 X10 because this is a linear combination.

01:37 This is the same as the sum of the expected values.

01:43 So it's expected value of x1 up to expected value of x10.

01:53 Now we have that all the morning week times are the same.

01:58 So they all have the value, expected value of x1.

02:02 So we have five times expected value of x1.

02:05 And all the evening times are the same.

02:06 They have the same expected value.

02:09 So we have five times the expected value of x6 for the evening times.

02:18 And we're given the expected values for the morning times is four.

02:24 So five times four.

02:26 And the expected value for the evening time is five.

02:32 So we have five times five.

02:35 So we have five times four plus five times five, which is 45.

02:45 So the expected total weight time is 10.

02:56 In part b, we're asked to find the variance of the total weighting time.

03:07 Well, we have the variance of x1 summed through x10.

03:16 This is the same as the sum of the variances, x1 up through the variance of x10.

03:26 And then because we have the evening times or independent of the morning waiting times, then this is simply what the variance is.

03:55 And so this can be simplified since the morning times have the same variance to five times the variance of x1, and the evening times also have the same variance, plus five times the variance of x6.

04:10 And we're given that the variance is for x1 and x6, these are calculated by squaring the length of the interval.

04:31 So we have 5 times 64 and dividing by 12.

04:56 And then we have squaring the other length...

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