MCQ+2 / -0.67MediumPMF, PDF & CDFProbabilityEngineering MathematicsMean, Median & ModeStatistics & RegressionStandard Deviation
Engineering Mathematics → Probability → Mean, Median & Mode
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Question
If f(x) and g(x) are two probability density functions,f(x)=⎩⎨⎧ax+1−ax+10:−a≤x<0:0≤x≤a:otherwiseg(x)=⎩⎨⎧−axax0:−a≤x<0:0≤x≤a:otherwiseWhich one of the following statements is true?
Correct answer
(B) Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different
Solution
1.Mean Analysis: Both f(x) and g(x) are symmetric about x=0 over the interval [−a,a]. For any probability density function symmetric about a point μ, the mean is μ. Therefore, E[X]f=0 and E[X]g=0. The means are the same.
2.Normalization: For f(x) and g(x) to be valid PDFs, the area under the curve must be 1.
∫−aaf(x)dx=a and ∫−aag(x)dx=a. This implies a=1 for them to be normalized PDFs.
3.Variance Analysis: Variance V[X]=E[X2]−(E[X])2. Since the means are 0, V[X]=E[X2].
For f(x) (assuming a=1):E[X2]f=∫−11x2f(x)dx=2∫01x2(1−x)dx=2[3x3−4x4]01=2(121)=61For g(x) (assuming a=1):E[X2]g=∫−11x2g(x)dx=2∫01x2(x)dx=2[4x4]01=21Since 61=21, the variances are different.Thus, the correct option is (B).
