Calculating f(x) PDF Mean,Median,Mode & Plotting

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In summary, to determine constants A and B for a valid PDF with a mean that is 0.9 times its mode, the function f(x) = Ax - Bx^3 is used. The mean and mode are calculated to be 1.1485 and 1.2761, respectively. The PDF is then plotted and the equations for mean and mode are set equal to solve for A and B. The integral of the function is also used to find another equation for A and B. Finally, the derivative of the function is taken to find the mode, and both equations are solved to determine the values of A and B.
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abhipatel
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Determine constants A and B such that function f(x) is a valid PDF whose mean is 0.9 times its mode.

f(x) = Ax- Bx ^3 , 0 <= x <= 2, otherwise = 0

Also calculate its mean,median and mode and plot the PDF

Please help...am scratching my head...have the answers but cannot derive the solution

Answers : A = 0.8466 ; B = 0.1733 ; Mean = 1.1485 , Median = 1.1724 , Mode = 1.2761
 
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  • #2
What are the definitions of mean and mode for a PDF? Set mean=mode*0.9. That gives you one equation in A and B. Now remember a PDF needs to have integral=1 over it's domain. That gives you another equation for A and B. Solve them.
 
  • #3
got one function...believe am screwing up in equating the mode and the mean i believe...mean is integral of x.f(x)dx right? basically integrate the function with limits 0 & 2...allright so u get 8A/3 - 32B/5 as Mean...Mode...am doing df(x)/dx...i don't know if that is correct?? thanks though am getting a good picture now...just a lil twist needed..
 
  • #4
got it my friend...excellent help..thanks a lot buddy..take care
 

1. How do I calculate the mean of a PDF for f(x)?

To calculate the mean of a PDF for f(x), you must first multiply each value of x by its corresponding probability, then add all of these products together. This sum is the mean of the PDF. Alternatively, you can use the formula: mean = ∑(x * P(x)), where P(x) is the probability of each value of x and the summation is taken over all values of x.

2. What is the median of a PDF for f(x)?

The median of a PDF for f(x) is the middle value when the data is arranged in ascending order. In other words, it is the value that divides the data into two equal halves. If there is an even number of data points, the median is the average of the two middle values. To find the median, you can also use the cumulative distribution function (CDF) to determine the value of x at which the CDF is equal to 0.5.

3. How do I find the mode of a PDF for f(x)?

The mode of a PDF for f(x) is the most frequently occurring value. To find the mode, you can either visually inspect the PDF plot and identify the highest peak, or you can use calculus to find the maximum point on the PDF curve. If there are multiple values with the same highest frequency, then the data is bimodal or multimodal, meaning there are multiple modes.

4. What is the difference between mean, median, and mode?

The mean, median, and mode are all measures of central tendency in a dataset. The mean is the average of all the data points, the median is the middle value, and the mode is the most frequently occurring value. The main difference between them is how they are affected by extreme values. The mean is influenced by outliers, while the median and mode are more robust to extreme values.

5. How can I plot a PDF for f(x)?

To plot a PDF for f(x), you must first calculate the probability of each value of x using the PDF function. Then, you can use a graphing tool or software to plot the values of x on the x-axis and their corresponding probabilities on the y-axis. The resulting graph will show the shape of the PDF curve and the location of the mean, median, and mode. Make sure to label your axes and provide a legend for clarity.

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