What makes a function of a discrete variable a candidate for a discrete random variable distribution? What about the counterpart of this candidacy in the case of a continuous variable? Explain in a one-page, APA formatted response.
Investment Classes Activity
For each investment class in Table 3, assume that future returns are normally distributed with the population mean and standard deviation as given. Based on this assumption:
For each investment class, find the probability of a return that is less than zero (that is, find the probability of a loss). Is your answer reasonable for all investment classes? Explain.
For each investment class, find the probability of a return that is:
Greater than 5%.
Greater than 10%.
Greater than 20%.
Greater than 50%.
For which investment classes is the probability of the return greater than 50% is essentially zero? For which investment classes is the probability of such a return greater than 1 percent? Greater than 5%?
For which investment classes is the probability of loss essentially zero?
For which investment classes is the probability of loss greater than 1%? Greater than 10%? Greater than 20%?
Full Answer Section
For a continuous variable, the counterpart to a discrete random variable distribution is a continuous probability distribution. Continuous probability distributions assign probabilities to a continuous range of values, rather than to individual points. The probability of a specific value is zero in a continuous distribution, but probabilities can be assigned to intervals of values.
Investment Class Analysis
Note: To provide accurate calculations, I would need the specific mean and standard deviation values for each investment class from Table 3. Assuming you have this data, here's a general approach:
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Calculate Probabilities:
- Use the normal distribution function (also known as the cumulative distribution function or CDF) to calculate the probabilities for each investment class and return threshold.
- For example, to find the probability of a return less than zero for a given investment class, you would calculate P(X < 0), where X is the random variable representing the return and the parameters of the normal distribution are the mean and standard deviation for that class.
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Interpret Results:
- Analyze the calculated probabilities to determine the likelihood of different return scenarios for each investment class.
- Consider whether the probabilities are reasonable based on your understanding of the investment classes and their historical performance.
- Identify investment classes with low probabilities of loss or high probabilities of returns above certain thresholds.
Example:
Assuming the mean return for a particular investment class is 5% and the standard deviation is 2%, you can calculate the probability of a return less than zero as follows:
- P(X < 0) = Φ(-5/2) ≈ 0.0062, where Φ is the standard normal distribution function.
This indicates a very low probability of a loss for this investment class.
By performing similar calculations for different return thresholds and investment classes, you can gain insights into the risk-return profiles of each option and make informed investment decisions.