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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)
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Solution
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`
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