Final answer:
The random variable X = -log U, where U follows a uniform distribution on [0,1], follows an exponential distribution with a mean (μ) and standard deviation (σ) of 1.
Step-by-step explanation:
If U follows a uniform distribution on [0,1], the random variable X = -log U does not follow a uniform distribution. Instead, X follows an exponential distribution with a rate parameter of 1. This result comes from the transformation of the uniform variable U through the negative logarithm function.
To elaborate, the probability density function (pdf) of U is f(u) = 1 for 0 ≤ u ≤ 1. When we apply the transformation X = -log U, we need to determine the new pdf for X. For an exponential distribution, the mean (μ) is the reciprocal of the rate parameter, and the standard deviation (σ) is also the reciprocal of the rate parameter. Since our rate is 1, both the mean and standard deviation of X will be 1.
Formulas for mean and standard deviation of X:
a. μ = 1/rate = 1
b. σ = 1/rate = 1
Conclusion:
Therefore, the distribution of X is exponential with μ = 1 and σ = 1, not uniform.