Question

Consider the continuous random variable X, which has a uniform distribution over the interval from 20 to 28.

Answer 1

Answer:

a) The probability distribution is $f(x) = \frac{1}{8}$

b) 0.5 = 50% probability that X will take on a value between 21 and 25.

c) 0.25 = 25% probability that X will take on a value of at least 26.

Step-by-step explanation:

Uniform probability distribution:

An uniform distribution has two bounds, a and b.

The probability of finding a value of at lower than x is:

$P(X < x) = \frac{a - x}{b - a}$

The probability of finding a value between c and d is:

$P(c \leq X \leq d) = \frac{d - c}{b - a}$

The probability of finding a value above x is:

$P(X > x) = \frac{b - x}{b - a}$

Uniform distribution over the interval from 20 to 28.

This means that $a = 20, b = 28$

a. What’s the probability density function?

The probability density function of the uniform distribution is:

$f(x) = \frac{1}{b - a}$

In this question:

$f(x) = \frac{1}{28 - 20} = \frac{1}{8}$

b. What’s the probability that X will take on a value between 21 and 25?

$P(21 \leq X \leq 25) = \frac{25 - 21}{28 - 20} = \frac{4}{8} = 0.5$

0.5 = 50% probability that X will take on a value between 21 and 25.

c. What’s the probability that X will take on a value of at least 26?

$P(X > 26) = \frac{28 - 26}{28 - 20} = \frac{2}{8} = 0.25$

0.25 = 25% probability that X will take on a value of at least 26.