Question

Let $X$ be a continuous random variable denoting the temperature measured. The range of temperature is $[0, 100]$ degree Celsius and the probability density function of $X$ be $f(x) = 0.01$ for $0 \le X \le 100$. The mean of $X$ is \(\underline{\hspace{2cm}}\)

• A. 2.5
• B. 5.0
• C. 25.0
• D. 50.0

Hint:

A constant PDF over an interval always represents a uniform distribution; its mean is simply the midpoint.

Answer 1

The Correct Option is D

Step 1: Identify the PDF.
The PDF is constant: \[ f(x) = 0.01, 0 \le x \le 100. \]

Step 2: Compute the mean of a uniform distribution.
A constant PDF over $[0,100]$ means $X$ is uniform on $[0,100]$. Mean of uniform distribution is: \[ E[X] = \frac{a + b}{2} = \frac{0 + 100}{2} = 50. \]

Step 3: Final conclusion.
Thus, the mean is: \[ \boxed{50} \]