Given that x is a random variable in the range [0, ∞) with a probability density function f(x) = ^-x/2K , the value of the constant K is: - Mathematics MCQ Mathematics Mock Test - 2 2026 | ExamTest.live | ExamTest.live

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Given that x is a random variable in the range [0, \infty) with a probability density function f (x) = \frac{e ^{- x /2}}{K}, the value of the constant K is:

A $2$
B $-7$
C $-2$
D $4$

Solution & Step-by-step Explanation

For a function to be a valid probability density function, the total area under the curve must be equal to 1 : \int_{0}^{\infty} f (x) d x = 1 \int_{0}^{\infty} \frac{e ^{- x /2}}{K} d x = 1 \frac{1}{K} \int_{0}^{\infty} e^{- x /2} d x = 1 \frac{1}{K} [\frac{e ^{- x /2}}{- 1/2}]_{0}^{\infty} = 1 \frac{1}{K} [- 2 e^{- x /2}]_{0}^{\infty} = 1 \frac{- 2}{K} [e^{- \infty} - e^{0}] = 1 \frac{- 2}{K} [0 - 1] = 1 \frac{2}{K} = 1 ⟹ K = 2

Practice this question

Try it yourself before checking the explanation above.

Given that x is a random variable in the range [0, \infty) with a probability density function f (x) = \frac{e ^{- x /2}}{K}, the value of the constant K is:

A

2

B

-7

C

-2

D

4

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