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Questions and Answers
What is the first step to compute the variance in the given text?
What is the first step to compute the variance in the given text?
- Extract the square root of the mean
- Calculate the square root of the result
- Subtract the square of the mean from the result (correct)
- Find the average of squared differences
What does the standard deviation of a probability distribution indicate?
What does the standard deviation of a probability distribution indicate?
- The square root of the variance
- The average of squared differences from the mean
- The sum of all probabilities
- The average of deviations from the mean (correct)
In Example 2, what random variable is used to construct the probability distribution?
In Example 2, what random variable is used to construct the probability distribution?
- Number of heads appearing (correct)
- Number of tails appearing
- Number of suits sold
- Number of cars sold
What does it mean to compute the variance of a probability distribution?
What does it mean to compute the variance of a probability distribution?
If a manager wants to ensure there are enough suits for the next 5 days, what should he purchase?
If a manager wants to ensure there are enough suits for the next 5 days, what should he purchase?
What does interpreting a variance entail in probability distributions?
What does interpreting a variance entail in probability distributions?
What is the expected value of the gain if you purchase one ticket?
What is the expected value of the gain if you purchase one ticket?
What does the negative expectation value mean in this context?
What does the negative expectation value mean in this context?
What does the variance describe in a distribution of a random variable?
What does the variance describe in a distribution of a random variable?
What is the net gain for a win when purchasing one ticket?
What is the net gain for a win when purchasing one ticket?
What is the average loss per ticket if you buy one ticket each week over a long time?
What is the average loss per ticket if you buy one ticket each week over a long time?
What does the standard deviation determine in a distribution of a random variable?
What does the standard deviation determine in a distribution of a random variable?
What does the mean of a random variable take into account when compared to the sample mean of a group of observations?
What does the mean of a random variable take into account when compared to the sample mean of a group of observations?
How is the mean of a discrete random variable calculated?
How is the mean of a discrete random variable calculated?
In the context of rolling a fair die, what does a theoretical mean of 3.5 imply?
In the context of rolling a fair die, what does a theoretical mean of 3.5 imply?
What does the random variable represent in the example of the security guard recording people entering the bank?
What does the random variable represent in the example of the security guard recording people entering the bank?
How is the expected value of a random variable different from its theoretical mean?
How is the expected value of a random variable different from its theoretical mean?
Why is it necessary to construct a probability distribution for a random variable when calculating its mean?
Why is it necessary to construct a probability distribution for a random variable when calculating its mean?
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Study Notes
Mean of a Discrete Random Variable
- The mean of a discrete random variable is the weighted average of its possible values, where each outcome is weighted according to its probability.
- The formula to compute the mean is:
- Steps to solve the mean of a discrete random variable:
- Construct the probability distribution for the random variable.
- Multiply the value of the random variable by the corresponding probability.
- Add the results obtained in Step 2.
Examples of Mean of a Discrete Random Variable
- Example 1: Rolling a fair die, the average number of spots that would appear is 3.5.
- Example 2: The average number of people entering the bank every hour during a working day is 2.5.
- Example 3: The expected value of the gain if you purchase one ticket for a color television valued at 35,000 pesos is -1,150 pesos.
Variance and Standard Deviation of a Discrete Random Variable
- The variance and standard deviation describe the amount of spread, dispersion, or variability of the items in a distribution.
- The formula to compute the variance is:
- The formula to compute the standard deviation is:
- Steps to solve the variance and standard deviation:
- Construct the probability distribution for the random variable.
- Multiply the value of the random variable by the corresponding probability.
- Add the results obtained in Step 2.
- Subtract the square of the mean from the result obtained in Step 4 to compute the variance.
- Extract the square root of the result to solve for the standard deviation.
Examples of Variance and Standard Deviation of a Discrete Random Variable
- Example 1: The variance and standard deviation of the probability distribution of the number of cars sold per day at a local car dealership are 0.80 and 0.89, respectively.
- Example 2: The variance and standard deviation of the probability distribution of the number of heads that appear when tossing 5 coins are 1.25 and 1.12, respectively.
- Example: The mean, variance, and standard deviation of the distribution of suits sold per day at a retail store are 2, 0.8, and 0.89, respectively.
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