Final answer:
To calculate the mean, variance, and standard deviation of a random variable X, one multiplies each possible value of X by its corresponding probability, sums these products for the mean, calculates weighted squared deviations for the variance, and takes the square root of the variance for the standard deviation.
Step-by-step explanation:
To solve these problems, one would typically use the probability distribution of the random variable X to compute various statistical measures. The mean or expected value of a discrete random variable is calculated by multiplying each possible value of the random variable by its probability and then summing all these products. The variance is found by taking the sum of the squared differences between each value and the mean, each weighted by its probability, and the standard deviation is the square root of the variance.
- Identify the possible values of the random variable X and their corresponding probabilities.
- Multiply each value of X by its probability to find the 'value times probability' for each outcome.
- Sum these products to find the mean of the distribution.
- For each value of X, subtract the mean and square the result, then multiply by its probability to find 'squared deviation times probability'.
- Sum these new products to calculate the variance.
- Take the square root of the variance to obtain the standard deviation.
The probability distribution of X, mean, variance, and standard deviation help to understand the behavior of the random variable X in the context of the examples provided. Graphs help visualize the distribution, while calculations provide numerical insights.