Let X be a continuous random variable denoting the temperature measured. The range of temperature is [0, 100] degree Celsius and let the probability density function of X be f(x) = 0.01 for 0 ≤ X ≤ 100.

The mean of X is ______

This question asks us to find the mean of a continuous random variable, denoted as X, which represents temperature. We are given the range of temperature as [0, 100] degrees Celsius and its probability density function (PDF), $f(x)=0.01$, for $0\le X\le 100$. This type of PDF, where the probability is constant over a specific interval, represents a uniform distribution.

Mean Calculation Formula for Continuous Variables

The mean, or expected value, $E(X)$ of a continuous random variable X with a probability density function $f(x)$ over the interval $[a,b]$ is calculated using the integral:

$$E(X)={\int }_a^{b}x\cdot f(x)dx$$

Applying the PDF and Range

In this specific problem:

- The lower limit of the range is $a=0$.
- The upper limit of the range is $b=100$.
- The probability density function is $f(x)=0.01$.

Substituting these values into the formula, we get:

$$E(X)={\int }_0^{100}x\cdot (0.01)dx$$

Step-by-Step Integration

To find the mean, we perform the integration:

1. Factor out the constant $0.01$: $$E(X)=0.01{\int }_0^{100}xdx$$
2. Integrate $x$ with respect to $x$. The integral of $x$ is $\frac{x^2}{2}$: $$E(X)=0.01{\left[\frac{x^2}{2}\right]}_0^{100}$$
3. Evaluate the definite integral using the limits $0$ and $100$: $$E(X)=0.01\left(\frac{(100{)}^2}{2}-\frac{(0{)}^2}{2}\right)$$
4. Calculate the values: $$E(X)=0.01\left(\frac{10000}{2}-0\right)$$ $$E(X)=0.01(5000)$$
5. Calculate the final result: $$E(X)=50.0$$

Verification of PDF

Before concluding, it's good practice to ensure the PDF is valid by checking if its integral over the entire range equals 1:

$${\int }_0^{100}f(x)dx={\int }_0^{100}0.01dx$$

$$=0.01[x{]}_0^{100}=0.01(100-0)=0.01\times 100=1$$

Since the integral equals 1, the PDF is valid.

Final Mean Value

The calculated mean of the continuous random variable X, representing the temperature, is 50.0 degrees Celsius.