Final answer:
Probabilities of the variable x taking on certain values are calculated using the distribution it follows, whether it's a normal distribution, a Poisson distribution, or a uniform distribution.
Step-by-step explanation:
The student's question requires calculating probabilities for the variable x taking on certain values. To do this, we must understand the type of distribution x follows and use the appropriate methods to calculate these probabilities.
For example, if x follows a normal distribution with a mean of 4 and a standard deviation of 5, we would use the normal distribution's cumulative distribution function (CDF) to find P(x < 4) and P(4 < x < 5).
In the case of a Poisson distribution, we can add up the probabilities of each whole number value of x that is less than or equal to 4 to find P(X ≤ 4). Similarly, for a uniform distribution, we can calculate the probability by finding the area under the density curve between the given values of x.
To find the maximum value of x in the bottom quartile for a continuous distribution, we would need to calculate the value of x where the CDF is 0.25.