Question

suppose the continuous random variable, x, is uniformly distributed between 1 and 10. what is the probability that x is either less than 4 or greater than 8? round to 2 decimal places.

Explanation:

Step1: Determine the probability density function

For a uniform distribution \(X\sim U(a = 1,b = 10)\), the probability density function \(f(x)=\frac{1}{b - a}=\frac{1}{10 - 1}=\frac{1}{9}\)

Step2: Calculate \(P(X\lt4)\)

Using the formula for the probability of a uniform distribution \(P(X\lt x)=\frac{x - a}{b - a}\), when \(x = 4\), \(P(X\lt4)=\frac{4 - 1}{10 - 1}=\frac{3}{9}\)

Step3: Calculate \(P(X\gt8)\)

Using the formula \(P(X\gt x)=\frac{b - x}{b - a}\), when \(x = 8\), \(P(X\gt8)=\frac{10 - 8}{10 - 1}=\frac{2}{9}\)

Step4: Calculate \(P((X\lt4)\cup(X\gt8))\)

Since the events \(X\lt4\) and \(X\gt8\) are mutually - exclusive, \(P((X\lt4)\cup(X\gt8))=P(X\lt4)+P(X\gt8)\)
\(P((X\lt4)\cup(X\gt8))=\frac{3 + 2}{9}=\frac{5}{9}\approx0.56\)

Answer:

\(0.56\)