The mean weight of luggage checked by a randomly selected...

The mean weight of luggage checked by a randomly selected tourist-class passenger flying between two cities on a certain airline is $40 \mathrm{lb},$ and the standard deviation is 10 $\mathrm{lb.}$ The mean and standard deviation for a business-class passenger are 30 lb and 6 lb, respectively. (a) If there are 12 business-class passengers and 50 tourist-class passengers on a particular flight, what are the expected value of total luggage weight and the standard deviation of total luggage weight? (b) If individual luggage weights are independent, normally distributed rvs, what is the probability that total luggage weight is at most 2500 $\mathrm{lb} ?$

The mean weight of luggage checked by a randomly selected tourist-class passenger flying
between two cities on a certain airline is $40 \mathrm{lb},$ and the standard deviation is 10 $\mathrm{lb.}$ The mean and standard deviation for a business-class passenger are 30 lb and 6 lb, respectively.
(a) If there are 12 business-class passengers and 50 tourist-class passengers on a particular
flight, what are the expected value of total luggage weight and the standard deviation of
total luggage weight?
(b) If individual luggage weights are independent, normally distributed rvs, what is the probability that total luggage weight is at most 2500 $\mathrm{lb} ?$

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is particularly useful in many practical situations due to the central limit theorem, which ensures that the sum of a large number of independent random variables tends towards a normal distribution, even if the original variables are not normally distributed.

Probability Using the Standard Normal Distribution

To find probabilities associated with normally distributed random variables, one often standardizes the variable to obtain a Z-score. This Z-score represents the number of standard deviations a value is from the mean, and it allows for the calculation of probabilities using standard normal distribution tables or software. This approach is used to determine the likelihood that the total luggage weight does not exceed a specified threshold.

Expected Value

The expected value, or mean, of a random variable is a measure of the central tendency. When combining multiple random variables, the linearity of expectation allows us to simply add their individual means, regardless of whether the variables are independent. This concept is crucial when determining the expected total luggage weight from different groups of passengers.

Variance and Standard Deviation of a Sum

For independent random variables, the variance of their sum is equal to the sum of their individual variances. The standard deviation, which measures the spread or variability around the mean, is the square root of the total variance. This principle is used to calculate the combined randomness in the total luggage weight when each passenger's luggage weight is considered independent.

Transcript

00:04 Told that the mean weight of luggage checked by a randomly selected passenger flying between two cities is 40 pounds and the standard deviation is 10 pounds we're told that the mean and standard deviation for a business class passenger on the other hand that was a tourist class for a business class class passenger are 30 pounds and 6 pounds in part a we're told that there are 12 business class and 50 tourist class passengers on a flight.

00:44 We're asked to find the expected value of the total luggage weight and the standard deviation of the total luggage weight.

00:57 Well, let x1 through x12 be the weights for the business class passengers.

01:19 And we'll let y1 through y50 be the tourist class weights.

01:36 Then the total weight t is going to be the sum over all the x's, and then the sum over all the y's as well, which we can actually write as the sum of two variables, x representing the total weight for the business class and y, representing the total weight for the tourist class.

02:09 We have that the expected value of x is going to be the same as 12 times the expected value of a single tourist in business class, x1.

02:20 And we're given that this is 30.

02:23 So this is going to be 12 times 30, which is 360 pounds.

02:32 Likewise, we have that the variance of x is going to be 12 squared times the variance of x1, which is going to be, oh, that's actually just, sorry, 12 times the variance of x1, since this is a linear combination.

02:59 And so this is 12 times 36, which is equal to 432.

03:16 The variance is found by squaring the standard deviation for x1.

03:21 We also have the expected value of y.

03:24 This is going to be 50 times the expected value of y1.

03:28 So 50 times we're told that the mean for the tourist class is 40, which gives us 2 ,000.

03:44 And likewise, the variance of y...

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