AAQIB IQBAL said:PROVE mean (X bar) of a continuous distribution is given by:
∫x.f(x)dx
{'a' is the lower limit of integration and 'b' is the upper limit}
The arithmetic mean, also known as the average, is the sum of all the values in a dataset divided by the number of values in the dataset. In continuous distributions, the arithmetic mean is calculated by integrating the product of each value and its corresponding probability density function.
The arithmetic mean of continuous distributions is calculated by finding the area under the curve of the probability density function and dividing it by the total area under the curve. This can also be expressed as the integral of the values multiplied by their respective probabilities, divided by the integral of the probability density function.
The arithmetic mean is a measure of central tendency that takes into account all values in a dataset, while the median is the middle value of a dataset. In continuous distributions, the arithmetic mean is influenced by extreme values, while the median is not. Additionally, the arithmetic mean can be calculated using mathematical formulas, while the median may need to be approximated using graphs or tables.
The arithmetic mean is important in continuous distributions because it is a useful measure of central tendency that allows us to summarize a large dataset into a single value. It can also be used to compare different datasets or to track changes in a dataset over time.
Yes, the arithmetic mean of continuous distributions can be negative if the dataset contains a mix of positive and negative values. However, the arithmetic mean is most commonly used with datasets that have only positive values. In these cases, a negative arithmetic mean may not be meaningful.