18 Questions
What is the first step to compute the variance in the given text?
Subtract the square of the mean from the result
What does the standard deviation of a probability distribution indicate?
The average of deviations from the mean
In Example 2, what random variable is used to construct the probability distribution?
Number of heads appearing
What does it mean to compute the variance of a probability distribution?
Calculate the sum of squared differences from the mean
If a manager wants to ensure there are enough suits for the next 5 days, what should he purchase?
Number equal to mean
What does interpreting a variance entail in probability distributions?
Understanding the average deviation from the mean
What is the expected value of the gain if you purchase one ticket?
₱0
What does the negative expectation value mean in this context?
You can win a television set valued at ₱35,000 or lose ₱100 on the ticket.
What does the variance describe in a distribution of a random variable?
The amount of spread or variability of the items in the distribution.
What is the net gain for a win when purchasing one ticket?
₱-35,000
What is the average loss per ticket if you buy one ticket each week over a long time?
₱0
What does the standard deviation determine in a distribution of a random variable?
The amount of spread or variability of the items in the distribution.
What does the mean of a random variable take into account when compared to the sample mean of a group of observations?
Probability of each outcome
How is the mean of a discrete random variable calculated?
Sum of products of values and corresponding probabilities
In the context of rolling a fair die, what does a theoretical mean of 3.5 imply?
Rolling the die many times will give an average close to 3.5
What does the random variable represent in the example of the security guard recording people entering the bank?
Number of people entering the bank
How is the expected value of a random variable different from its theoretical mean?
Expected value considers all possible outcomes, theoretical mean doesn't
Why is it necessary to construct a probability distribution for a random variable when calculating its mean?
To understand how likely each outcome is
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.
Learn how to calculate the mean or expected value of a discrete random variable by weighting each outcome according to its probability. Understand the formula and steps involved in solving the mean of a discrete random variable.
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