Questions: A cumulative distribution function for a continuous random variable, X, is given by: F(x)=(x-1)/80, for 1 ≤ x ≤ 3 a) Find P(X ≤ 2). Round your answer to two decimal places. b) What is P(1.5 ≤ X ≤ 2.5) ? Round your answer to two decimal places.

Solution Steps

To solve these problems, we will use the properties of the cumulative distribution function (CDF). The CDF, \( F(x) \), gives the probability that the random variable \( X \) is less than or equal to \( x \).

a) To find \( P(X \leq 2) \), we simply evaluate the CDF at \( x = 2 \).

b) To find \( P(1.5 \leq X \leq 2.5) \), we calculate the difference \( F(2.5) - F(1.5) \).

To find \( P(X \leq 2) \), we evaluate the cumulative distribution function (CDF) at \( x = 2 \):

\[ F(2) = \frac{2 - 1}{80} = \frac{1}{80} = 0.0125 \]

Rounding to two decimal places, we have:

\[ P(X \leq 2) = 0.01 \]

To find \( P(1.5 \leq X \leq 2.5) \), we calculate the difference \( F(2.5) - F(1.5) \):

\[ F(2.5) = \frac{2.5 - 1}{80} = \frac{1.5}{80} = 0.01875 \]

\[ F(1.5) = \frac{1.5 - 1}{80} = \frac{0.5}{80} = 0.00625 \]

\[ P(1.5 \leq X \leq 2.5) = F(2.5) - F(1.5) = 0.01875 - 0.00625 = 0.0125 \]

Rounding to two decimal places, we have:

\[ P(1.5 \leq X \leq 2.5) = 0.01 \]

Final Answer