Final answer:
The probability for a continuous random variable from a probability density function is the area under the curve between two points. The mean is the integral of x times the pdf, and the standard deviation is the square root of the variance, which is the integral of the squared difference between x and the mean times the pdf.
Step-by-step explanation:
To find the probability for a continuous random variable described by a probability density function (pdf), we must calculate the area under the curve defined by the function f(x). This requires the boundaries (a and b) within which we're interested in finding the probability.
To calculate the probability that X falls within a specific range, we integrate the pdf over that range, that is P(a ≤ x ≤ b) = ∫ f(x) dx where the limits of integration are a and b.
Finding the Mean and Standard Deviation of a Continuous Random Variable
The mean (expected value) of a continuous random variable X is found by calculating the integral of x multiplied by the pdf over the entire range of X. Similarly, the variance is the integral of the square of the difference between x and the mean, again multiplied by the pdf. The standard deviation is then the square root of the variance:
Mean (μ) = ∫ x f(x) dx
Variance (σ²) = ∫ (x - μ)² f(x) dx
Standard Deviation (σ) = √(σ²)
To answer specific questions regarding probability, the mean, or the standard deviation, the exact form of the pdf f(x) and the range of the variable X would be essential.