Final answer:
The question revolves around computing the probability of a range for a random variable and finding its mean and standard deviation, requiring integration over the specified range given the probability density function.
Step-by-step explanation:
The student's question concerns determining the probability of an event and finding the mean and standard deviation of a random variable X. For a continuous random variable X in a range (a, b), the probability that X is between any two values (c, d) is found by computing the area under the probability density function between those two points.
To find the mean (i) and the standard deviation (ii) of X, one would need to integrate the function f(x) multiplied by x over the interval (a, b) to find the mean, and similarly use the definition of variance and then take the square root for the standard deviation. However, without a specific function provided, we can't calculate these exactly.
When dealing with independent random variables Y and Z, the combined probability can be found through multiplication if we consider the joint occurrence ('AND' case) or through addition-subtraction formula if we consider either event occurring ('OR' case).