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.

Let us understand the question and calculate the mean of the continuous random variable X step-by-step.

We are given that X is a continuous random variable representing the measured temperature. The range of temperature is [0, 100] degrees Celsius, and its probability density function (PDF) is given by:

$f(x)=0.01\text{for}0\le x\le 100$

Since the probability density function is constant over the interval [0, 100], X follows a continuous uniform distribution, denoted as U(a, b), where the lower limit a = 0 and the upper limit b = 100.

The mean (or expected value) of a continuous random variable X with a probability density function f(x) is defined by the formula:

$E[X]={\int }_{-\infty }^{\infty }xf(x)dx$

Since f(x) is non-zero only in the interval [0, 100], we can restrict the integration limits to this interval:

$E[X]={\int }_0^{100}x\cdot 0.01dx$

We can factor out the constant 0.01 from the integral:

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

Now, evaluate the integral of x, which is equal to x² / 2:

$E[X]=0.01{\left[\frac{x^2}{2}\right]}_0^{100}$

Substitute the upper and lower limits of integration:

$E[X]=0.01(\frac{{100}^2}{2}-\frac{0^2}{2})$

$E[X]=0.01\left(\frac{10000}{2}\right)$

$E[X]=0.01\cdot 5000$

$E[X]=50.0$

Alternatively, since X is uniformly distributed on the interval [a, b], we can use the formula for the mean of a uniform distribution:
Mean = (a + b) / 2
Mean = (0 + 100) / 2 = 50.0.

Therefore, the mean of X is indeed 50.0.