Answer:
a) 0.0278
b) 0.5556
Step-by-step explanation:
We are given the following in the question:
The probability density function for a random variable X is given by
[tex]f(x) = \displaystyle\frac{x}{18}\\\\0 < x < 6[/tex]
We can find the probabilities as:
[tex]P(X<c) = \displaystyle\int_{-\infty}^c f(x) dx\\\\\\P(X>c) = \displaystyle\int^{\infty}_c f(x) dx[/tex]
a) P(X < 1)
[tex]P(X<1) = \displaystyle\int_{-\infty}^1 f(x) dx\\\\P(X<1) =\displaystyle\int_{0}^1 \frac{x}{18} dx\\\\= \frac{1}{18}\displaystyle\int_{0}^1 x~ dx = \frac{1}{18}\Big[\frac{x^2}{2}\Big]^1_0\\\\= \frac{1}{18}\Big(\frac{1}{2}-0\Big) = \frac{1}{36} = 0.0278[/tex]
b) P(X > 4)
[tex]P(X>4) = \displaystyle\int^{\infty}_4 f(x) dx\\\\P(X>4) = \displaystyle\int^{6}_4 \frac{x}{18} dx\\\\= \frac{1}{18}\displaystyle\int_{4}^6 x~ dx = \frac{1}{18}\Big[\frac{x^2}{2}\Big]^6_4\\\\= \frac{1}{18}\Big(\frac{36}{2}-\frac{16}{2}\Big) = \frac{1}{36}\Big(10\Big) = \frac{10}{18} = 0.5556[/tex]
