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प्रश्न
Find k if the following function represent p.d.f. of r.v. X
f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P `(1/ 4 < x < 3 /2)`.
उत्तर
Since, the function f is p.d.f. of X
∴` int_(-∞)^∞ f (x) dx` = 1
∴` int_(-∞)^0 f (x) dx`+ ` int_(0)^2 f (x) dx` + ` int_(2)^∞ f (x) dx` = 1
∴ 0 + ` int_(0)^2 kxdx` + 0 =1
∴ `k [x^2/2]_0^2` =1
∴ `k [4/2 - 0] =1`
∴ 2k = 1
∴ k =`1/2`
P `(1/ 4 < x < 3 /2)` = ` int_(1/4)^(3/2) f (x) dx`
` int_(1/4)^(3/2) kxdx`,
where k= `1/2`
= `1/2 int_(1/4)^(3/2) x dx`
= `1/2[x^2/2]_(1/4)^(3/2)`
= `1/4[9/4-1/16]`
= `1/4[(36-1)/16]`
=`35/64`
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