## Related questions with answers

Question

Graph the density function f(x)=kx²(0≤x≤5; k suitable) and the distribution function.

Solution

VerifiedStep 1

1 of 3We must have

$\int_{-\infty}^{\infty} f(x) \, dx = 1$

Since $f(x) = 0$ when $x < 0$ or $x > 5$,

$\int_{-\infty}^{\infty} f(x) \, dx = \int_{0}^5 f(x) \, dx = \int_0^5 kx^2 \, dx = \dfrac{kx^3}{3} \biggr \rvert_0^5 = \dfrac{5^3 k}{3} - \dfrac{0^3k}{3} = \dfrac{125}{3}k$

Thus,

$\dfrac{125}{3}k = 1 \Longrightarrow \boxed{\color{#4257b2}k = \dfrac{3}{125}}$

This means that

$f(x) = \begin{cases} \dfrac{3}{125}x^2, & 0 \leqslant x \leqslant 5 \\ 0 & \text{otherwise} \end{cases}$

Graph this function:

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