Question

Suppose X has a continuous uniform distribution over the interval
[2.4, 5.4].

What is P(X < 4.2)

Answer 1

Since it is uniform, f(x) = k
Solve:
$$ \Large \int_{2.4}^{5.4} k ~dx= 1

$$

Answer 2

the area under the curve must equal to 1 , over the domain

Answer 3

makes sense, yes.

Answer 4

In general :
For a uniform distribution f(x) = k defined over interval [a,b]
$$ \Large{ \int_{a}^{b} k ~dx= 1
\\ ~\\
\left [~kx~ \right ]_{b}^{a}= 1
\\ ~\\
k(b-a) = 1

\\ ~\\
k = \frac{1}{b-a}

}
$$

Answer 5

therefore
f(x) = 1 / ( 5.4 - 2.4) = 1/3

Answer 6

$$
\Large {
P(X<4.2) = \int_{2.4}^{4.2} \frac1 3 ~dx


}

$$

Answer 7

1/3(4.2-2.4) = 1/3 (1.8) =0.6

Answer 8

Thank you very much !!!!!!!! :D

Answer 9

theres a small typo above