Question

Consider a random variable X having the uniform density function f ( x ) with a = 20 and b= 30
i) Define and graph the density function f(x).
ii) Verify that f(x) is a probability density function.
iii) Find P(22≤x≤30). iv) Find p( x= 25)

Answer 1

Answer on Question#65392 – Math – Statistics and Probability

**Question.** Consider a random variable $X$ having the uniform density function $f(x)$ with $a = 20$ and $b = 30$.

i) Define and graph the density function $f(x)$.

**Solution.** Using the definition of uniform continuous density function (see https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)) we obtain:

$$f(x) = \begin{cases} \frac{1}{30-20}, & 20 \leq x \leq 30 \\ 0, & x < 20 \text{ or } x > 30 \end{cases} = \begin{cases} \frac{1}{10}, & 20 \leq x \leq 30 \\ 0, & x < 20 \text{ or } x > 30 \end{cases}$$

**Answer.** $f(x) = \begin{cases}

\frac{1}{10}, & 20 \leq x \leq 30 \\

0, & x < 20 \text{ or } x > 30

\end{cases}$

ii) Verify that $f(x)$ is a probability density function.

**Proof.** Obviously $f(x) \geq 0$ for all $x \in \mathbb{R}$. Now we must check that $\int_{-\infty}^{\infty} f(x) dx = 1$ (see https://onlinecourses.science.psu.edu/stat414/node/97, Definition).

Obviously $\int_{-\infty}^{20} 0 dx = \int_{30}^{\infty} 0 dx = 0$. Then

$\int_{-\infty}^{\infty} f(x) dx = \int_{20}^{30} \frac{1}{10} dx$. Using the linearity and Second fundamental theorem of calculus

(see https://en.wikipedia.org/wiki/Integral) we get:

$$\int_{20}^{30} \frac{1}{10} dx = \frac{1}{10} \int_{20}^{30} dx = \frac{1}{10} x \big|_{x=20}^{30} = \frac{1}{10} \cdot (30 - 20) = \frac{1}{10} \cdot 10 = 1.$$

Indeed $f(x)$ is a probability density function.

iii) Find $P(22 \leq X \leq 30)$.

**Solution.** Using https://onlinecourses.science.psu.edu/stat414/node/97, Definition, (3) we get:

$$P(22 \leq X \leq 30) = \int_{22}^{30} f(x) dx = \int_{22}^{30} \frac{1}{10} dx = \frac{1}{10} \cdot (30 - 22) = \frac{8}{10} = \frac{4}{5} = 0.8.$$

Answer. 0.8.

iv) Find $P(X = 25)$.

**Solution.** Similarly to https://onlinecourses.science.psu.edu/stat414/node/97 (see example with $P\big(X = \frac{1}{2}\big)$) we have:

$$P(X = 25) = \int_{25}^{25} \frac{1}{10} dx = \frac{1}{10} \cdot (25 - 25) = \frac{1}{10} \cdot 0 = 0.$$

Answer. 0.

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