Consider a random variable x that takes values +1 and -1 with probabil...
The PDF of the random variable is
The cumulative distribution function F(x) is the probability upto x as given below
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Consider a random variable x that takes values +1 and -1 with probabil...
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) of a random variable gives the probability that the random variable takes on a value less than or equal to a given value. In other words, it gives the cumulative probability distribution of the random variable.
Given Information
The given random variable x takes values 1 and -1 with probability 0.5 each. This means that the probability of x being equal to 1 is 0.5, and the probability of x being equal to -1 is also 0.5.
Determining the CDF at x = -1 and x = 1
To determine the CDF at x = -1 and x = 1, we need to calculate the cumulative probabilities.
At x = -1:
The cumulative probability at x = -1 is the probability that x takes on a value less than or equal to -1. Since the only possible value of x that is less than or equal to -1 is -1 itself, the cumulative probability at x = -1 is equal to the probability of x being -1. In this case, the probability of x being -1 is 0.5. Therefore, the CDF at x = -1 is 0.5.
At x = 1:
Similarly, the cumulative probability at x = 1 is the probability that x takes on a value less than or equal to 1. Since the only possible value of x that is less than or equal to 1 is 1 itself, the cumulative probability at x = 1 is equal to the probability of x being 1. In this case, the probability of x being 1 is also 0.5. Therefore, the CDF at x = 1 is 0.5.
Final Answer
The value of the cumulative distribution function F(x) at x = -1 is 0.5, and at x = 1 is also 0.5. Therefore, the correct answer is option 'C' - 0 and 1.
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