Question

It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = `x^2 /3` , for –1 < x < 2 and = 0 otherwise

 Verify whether f (x) is p.d.f. of r.v. X.

Answer 1

f (x) = `x^2/3 ≥0,` for -1 < x < 2

Also, ` int_(-∞)^∞ f (x) dx`

=`int_(-∞)^-1 f (x) dx`+ `int_(-1)^2 f (x) dx`  +`int_(2)^∞f (x) dx` 

= 0+`int_(-1)^2 f (x^2/3) dx` + 0 = `1/3[x^3/3]_-1^2`

= `1/3[8/3 - ((-1))/3] = 1/3[9/3] = 1`

∴ f (x) is the p.d.f. of X.