Determine whether the function is a probability density function over the given interval.
-$f ( x ) = \frac { 1 } { 8 } x , \quad 0 \leq x \leq 4$

Question

Quizplus · Accepted Answer

To be a probability density function over the given interval, the function must satisfy two conditions: (1) it must always be non-negative, and (2) its integral over the interval must equal 1.

In this case, we can check that f(x) is always non-negative, since x is non-negative and the fraction 1/8 is positive.

To check the second condition, we integrate f(x) over the interval [0,4]:
∫[0,4] f(x) dx = ∫[0,4] (1/8)x dx = (1/8) * (1/2) * x^2 |[0,4] = (1/16) * 4^2 - (1/16) * 0^2 = 1

Since the integral of f(x) over the interval [0,4] equals 1, f(x) is indeed a probability density function over the given interval. Therefore, the answer is A.

Determine Whether the Function Is a Probability Density Function Over f(x)=18x,0≤x≤4f ( x ) = \frac { 1 } { 8 } x , \quad 0 \leq x \leq 4f(x)=81​x,0≤x≤4

Determine Whether the Function Is a Probability Density Function Over $f ( x ) = \frac { 1 } { 8 } x , \quad 0 \leq x \leq 4$