The probability that a normal random variable is less than its mean is ______.Multiple ChoiceCannot be determined0.01.00.5

Break Down the Problem

1. Identify the nature of the random variable: It is a normal random variable.
2. Understand the properties of the normal distribution: The mean of a normal distribution divides the distribution into two equal halves.

Relevant Concepts

1. For a normal distribution, the probability density function is symmetric about the mean.
2. The cumulative distribution function (CDF) of a normal random variable at its mean is equal to 0.5.

Analysis and Detail

1. Since the normal distribution is symmetric, the area under the curve to the left of the mean is equal to the area to the right of the mean.
2. Therefore, the probability that the normal random variable is less than its mean is:

$P(X < \mu) = 0.5$

Verify and Summarize

1. The normal distribution’s properties guarantee that 50% of the values fall below the mean and 50% fall above it.
2. Hence, the probability that a normal random variable is less than its mean is confirmed to be 0.5.

Final Answer

The probability that a normal random variable is less than its mean is 0.5.