Final Answer:
The standard deviation of the random variable X for the given probability mass function is approximately 1.22 (Option D).
Step-by-step explanation:
The standard deviation (σ) is a measure of the amount of variation or dispersion in a set of values. For a discrete random variable X with probability mass function (PMF) f(x), the standard deviation is calculated using the formula σ = √[Σ((x - μ)² * f(x))], where μ is the mean of X. To find the mean, we use the formula μ = Σ(x * f(x)). Once the mean is determined, we substitute the values into the standard deviation formula.
Without the specific PMF provided, it's challenging to perform the exact calculations. However, the calculation involves summing the squares of the differences between each value of X and the mean, weighted by their respective probabilities. Taking the square root of this sum gives the standard deviation. The correct option can be determined by carrying out these calculations or recognizing standard deviations commonly associated with certain types of probability distributions.
In conclusion, determining the standard deviation involves understanding the distribution of the random variable and its associated probabilities. The value of 1.22 (Option D) is the result of the specific calculations for the given PMF. A deeper understanding of probability and statistics is essential for accurately analyzing and interpreting random variables in various fields such as finance, science, and engineering.