Question

If f(x) and g(x) are two probability density functions,f(x)={xa+1:−a≤x<0−xa+10≤x≤a0 otherwise g(x)={−xa:−a≤x≤0xa:0≤x≤a0: otherewise Which one of the following statements is true?

• A. (A) Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are same
• B. (B) Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different
• C. (C) Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same
• D. (D) Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different

Answer 1

Answer: (B)

(B) Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different

Answer 2

Explanation:
Mean of f(x):E(x)=∫−a0X(Xa+1)dx+∫0aX(−Xa+1)dx=(X33a+X22)−a0+(−X33a+X33)0a=0=∫−a0X2(Xa+1)dx+∫0aX2(−Xa+1)dx(X44a+X33)−a0+(−X44a+X33)0a=a36⇒ Variance =a36Mean of g(x):E(x)=∫−a0x(−xa)dx+∫0ax×(Xa)dx=0Variance of g(x) is E(x2)−{E(X)}2, Where E(X2)=∫−a0X2(−Xa)dX+∫0aX2(Xa)dx=a32⇒ Variance =a32∴ Mean of f(x) and g(x) are same but variance of f(x) and g(x) are different.Hence, the correct option is (B).