Question

Consider a continuous random variable \(X\) which follows a normal distribution. A sample of size \(n = 25\) is drawn from the distribution of \(X\). Sample mean is 5. Sample standard deviation is 1.5. The probability of \(P[X = 5]\) is _____ (Rounded off to two decimal places).

Hint:

Remember that for any continuous random variable, the probability of it taking one exact value is always zero, regardless of the mean or standard deviation.

Answer 1

Correct Answer: 0

Step 1: Recall the defining property of a continuous random variable.
For a continuous random variable \(X\) with probability density function \(f(x)\), the probability over an interval \([a, b]\) is the area under \(f(x)\) between \(a\) and \(b\), that is \(P(a \le X \le b) = \int_a^b f(x)\, dx\).
Step 2: Apply this to a single point \(X = 5\).
Asking for \(P[X = 5]\) is the same as asking for \(P(5 \le X \le 5)\), which is the area under \(f(x)\) over an interval of zero width. Since \[ P(5 \le X \le 5) = \int_5^5 f(x)\, dx = 0 \] this integral is always zero, no matter what \(f(x)\) looks like.
Step 3: Note that the given sample statistics are not needed.
The sample size \(n = 25\), sample mean 5, and sample standard deviation 1.5 would be needed to estimate the population mean and variance of \(X\) if we wanted, say, \(P(X \le 5)\) or \(P(4 \le X \le 6)\) using the normal CDF, but they play no role in the probability of a single exact point because that probability is identically zero for every continuous distribution. \[ \boxed{P[X = 5] = 0.00} \]