Mean of a probability distribution

In summary, the conversation discusses finding the mean of a probability distribution, using equations such as the integral of a probability distribution and the mean squared. The solution is found to be c=1/b, with a final calculation error leading to the incorrect answer of x-bar=0.
  • #1
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[SOLVED]Mean of a probability distribution

Homework Statement


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Homework Equations


[tex]\int^{b}_{a}p(x)dx=1[/tex]

[tex]V=M_2-\bar{x}^2[/tex]

[tex]\bar{x}^2=\int^{b}_{a}xp(x)dx[/tex]

[tex]M_2=\int^{b}_{a} x^2p(x)dx[/tex]

The Attempt at a Solution



I found that [tex]c=\frac{1}{b}[/tex] which is a right answer.

What I did next was:

[tex]
\bar{x}=\int^{b}_{-b}xp(x)dx[/tex]

[tex]=\int^{0}_{-b}x(\frac{cx}{b}+c)dx\ + \int^{b}_{0}x(\frac{-cx}{b}+c)dx[/tex]

[tex]= \int^{0}_{-b}\frac{cx^2}{b}\ +\ c\ dx\ + \int^{b}_{0}\frac{-cx^2}{b}\ +\ c\ dx[/tex]

[tex]=\left[ \frac{cx^3}{3b}+cx \right]_{-b}^{0}+\left[ \frac{-cx^3}{3b}+cx \right]_{0}^{b}[/tex]

[tex]=\frac{-2cb^3}{3b}+{2cb}[/tex]

[tex]=\frac{-2b^2}{3b}+\frac{2b}{b}[/tex]

[tex]=\frac{-2b}{3}+2
[/tex]

But the answer says that [tex]\bar{x}=0[/tex]

If I can manage to get x-bar, I can manage to get the variance and SD.
 
Last edited:
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  • #2
i) you are integrating cx, that should give you an antiderivative of cx^2/2. ii) put in the top limit and subtract the bottom limit. You are making sign errors. The cx^3/3b terms also cancel.
 
  • #3
What an amateur mistake. I feel foolish. Cheers, Dick.
 

1. What is the definition of the mean of a probability distribution?

The mean of a probability distribution, also known as the expected value, is a measure of central tendency that represents the average value of the data in the distribution. It is calculated by multiplying each possible outcome by its probability of occurrence and summing the results.

2. How is the mean of a probability distribution different from the mean of a sample or population?

The mean of a probability distribution is a theoretical value that represents the average outcome of an infinite number of experiments. In contrast, the mean of a sample or population is a descriptive statistic that represents the average of the observed data points. The mean of a probability distribution is also calculated using probabilities, while the mean of a sample or population is calculated using actual data values.

3. What is the relationship between the mean and the variance of a probability distribution?

The mean and variance of a probability distribution are both measures of central tendency, but they represent different aspects of the distribution. The mean represents the average value, while the variance represents the spread or variability of the data. In some cases, a higher mean may be accompanied by a higher variance, indicating a wider range of possible outcomes.

4. How is the mean used to make predictions about the probability distribution?

The mean of a probability distribution can be used to make predictions about the likelihood of certain outcomes occurring. For example, if the mean is higher, it is more likely that the outcomes will be closer to that value. However, the mean alone does not provide a complete picture of the distribution and should be considered along with other measures, such as the variance or standard deviation.

5. How can the mean of a probability distribution be affected by outliers?

Outliers, or extreme values, can greatly influence the mean of a probability distribution. If there are a few outliers with very high or low values, the mean may be skewed towards one end of the distribution. In this case, it may be more appropriate to use a different measure of central tendency, such as the median, which is less affected by extreme values.

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