Consider a random variable X that has the probability of 0.5 foreach of the value +1 and -1. At X=-1 and +1, the values of the cumulative distribution function F(x) are 0 and 0.5 0 and 1 0.5 and 1 0.25 and 0.75
Question
Consider a random variable X that has the probability of 0.5 foreach of the value +1 and -1. At X=-1 and +1, the values of the cumulative distribution function F(x) are 0 and 0.5 0 and 1 0.5 and 1 0.25 and 0.75
Solution
The cumulative distribution function (CDF) for a random variable is defined as the probability that the variable takes a value less than or equal to a certain value.
In this case, the random variable X can take the values -1 and +1, each with a probability of 0.5.
The CDF at X=-1 is the probability that X takes a value less than or equal to -1. Since -1 is the smallest possible value for X, this is simply the probability that X=-1, which is 0.5.
The CDF at X=+1 is the probability that X takes a value less than or equal to +1. Since +1 is the largest possible value for X, this is the sum of the probabilities that X=-1 and X=+1, which is 0.5 + 0.5 = 1.
So, the values of the cumulative distribution function F(x) at X=-1 and +1 are 0.5 and 1, respectively.
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