Final answer:
The mean of the given probability distribution is 2.5. The correct variance and standard deviation cannot be confirmed with the options provided as they don't match the theoretically calculated standard deviation of √1.25 from the reference material.
Step-by-step explanation:
To find the mean of a given probability distribution, you multiply each possible outcome by its probability and sum all these products together. Based on the provided theoretical mean calculation, we use the formula: Mean (μ) = ∑(xi * P(xi)), which results in a mean of 2.5. This aligns with option (a).
To calculate the variance, we apply the formula: Variance (σ2) = ∑((xi - μ)2 * P(xi)). Given the theoretical standard deviation is the square root of 1.25, the variance is 1.25; however, since this value is not among the options, we might need additional context or data to provide a definitive answer from the provided options.
The standard deviation is the square root of the variance, and since it is given as √1.25, this corresponds to approximately 1.12, which does not match any of the provided options. It seems there could be a discrepancy that needs clarification since none of the options listed (1.41, 1.73, 2.00, 2.24) correspond to the calculated standard deviation of √1.25.