Let X is 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 __________.

The correct option is 50.0.

Step-by-Step Explanation:
We are given a continuous random variable $X$ representing the temperature measured in the interval $[0,100]$ degrees Celsius.
The probability density function (PDF) of $X$ is given by:

$f\left(x\right)=0.01$ for $0\le x\le 100$

The mean (or expected value) of a continuous random variable $X$ with PDF $f\left(x\right)$ defined over an interval $[a,b]$ is calculated using the formula:

$E\left[X\right]={\int }_a^{b}x·f\left(x\right)dx$

Substituting the limits $a=0$, $b=100$, and $f\left(x\right)=0.01$ into the formula, we get:

$E\left[X\right]={\int }_0^{100}x·\left(0.01\right)dx$

Factor out the constant $0.01$:

$E\left[X\right]=0.01{\int }_0^{100}xdx$

Now, integrate $x$ with respect to $x$:

${\int }_0^{100}xdx={\left[\frac{x^2}{2}\right]}_0^{100}$

Evaluate the definite integral at the upper and lower limits:

${\left[\frac{x^2}{2}\right]}_0^{100}=\frac{{100}^2}{2}-\frac{0^2}{2}=\frac{10000}{2}=5000$

Multiply by the constant factor $0.01$:

$E\left[X\right]=0.01\times 5000=50.0$

Thus, the mean of $X$ is 50.0.