Final answer:
To calculate the variance of the random variable X, the specific probability distribution of X must be known. Variance calculations differ based on whether the distribution is uniform, normal, exponential, or another type. Without this information, it is not possible to accurately determine the variance from the provided options.
Step-by-step explanation:
To determine the variance of a random variable X, we need to know the probability distribution of X. A probability distribution describes how probabilities are distributed over the values of the random variable. For example, if X is uniformly distributed, then the variance would be calculated differently compared to a normal distribution or an exponential distribution.
- For a uniform distribution U(a, b), the variance is given by σ² = (b-a)²/12.
- For a normal distribution with mean μ and standard deviation σ, the variance is simply σ² (the square of the standard deviation).
- For an exponential distribution with rate λ, the variance is 1/λ².
The student's question has left out the specific probability distribution of X, which is critical to answering the question accurately. Given options 1) 1.0 2) 5.0 3) 2.25 4) 2.24, these numbers may represent possible values for the variance but without the distribution's form, it's not possible to select the correct variance.
To sketch a graph of the probability distribution, calculate mean and standard deviation, and find the probability for certain scenarios, one would need additional information such as the parameters of the distribution (i.e., mean, standard deviation, rate) or the nature of the distribution given its type (exponential, normal, uniform, etc.).