A continuous random variable X has the p.d.f., f (x) = 3x^2; 0 ≤ x ≤ 1. The value of a constant λ that satisfies the relation Pr {X ≤ λ} = Pr {X > λ} is ? a) (1/3)^1/2 b) (1/2)^1/3 c) (2/3)^1/2 d) (2/3)^1/3
1 Answer
lamda = root(3)(1/2)
Or, Equivalently,
Explanation:
If
int_(-oo)^(oo) \ f(x) \ dx = 1
We note that:
int_(0)^(1) \ 3x^2 \ dx = [x^3]_0^1 = 1
Hence, the complete probability density function is:
f(x) = { (3x^2, 0 le x le 1), (0, "otherwise") :}
So we seek the value of
P(X le lamda) = P(X gt lamda )
As
P(X le lamda) = P(X ge lamda )
=> P(X le lamda) = 1 - P(X le lamda )
=> 2P(X le lamda) = 1
:. P(X le lamda) = 1/2
Thus we require that:
int_(0)^(lamda) \ 3x^2 \ dx = 1/2
=> [x^3]_(0)^(lamda) = 1/2
:. lamda^3 - 0 = 1/2
:. lamda^3 = 1/2
:. lamda = root(3)(1/2)
Or, Equivalently,