Question

The p.d.f. of X is defined as
f(x) = `{{:("k"",",  "for"  0 < x ≤ 4),(0",",  "otherwise"):}`
Find the value of k and also find P(2 ≤ X ≤ 4)

Answer 1

Let X and a random variable if a Probability density function

`int_(-oo)^oo "f"(x)  "d"x` = 1

Here `int_0^4  "f"(x)  "d"x` = 1

`int_0^4 "k"  "d"x` = 1

⇒ `"k"[x]_0^4` = 1

`"k"[4 - 0]` = 1

⇒ 4k = 1

∴ k = `1/4`

P(2 ≤ x ≤ 4) = `int_2^4 "f"(x) "d"`

=  `int_2^4 "kd"x`

= `int_2^4 1/4  "d"x`

= `1/4 int_2^4 "d"x`

= `1/4 [x]_2^4`

= `1/4 [4 - 2]`

= `1/4 [2]`

= `1/2`