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...
The cumulative distribution function (CDF) gives the probability that a random variable takes on a value less than or equal to a given value. In this case, the random variable x takes values of 1 and -1 with equal probability of 0.5 each. Let's calculate the CDF for both values.
Calculating CDF for x = -1:
The CDF at x = -1, denoted as F(-1), is the probability that x takes on a value less than or equal to -1. Since x can only take on the values 1 and -1, we need to calculate the probability that x is less than or equal to -1, which is P(x ≤ -1).
Since x can only take on the values 1 and -1 with equal probability, the probability that x is less than or equal to -1 is equal to the probability that x is equal to -1, which is 0.5.
Therefore, F(-1) = P(x ≤ -1) = 0.5.
Calculating CDF for x = 1:
Similarly, the CDF at x = 1, denoted as F(1), is the probability that x takes on a value less than or equal to 1. We need to calculate P(x ≤ 1).
Since x can only take on the values 1 and -1 with equal probability, the probability that x is less than or equal to 1 is equal to the probability that x is equal to 1, which is also 0.5.
Therefore, F(1) = P(x ≤ 1) = 0.5.
Conclusion:
The CDF for x = -1 is 0.5, and the CDF for x = 1 is also 0.5. Therefore, the correct answer is option 'C', which states that the value of the cumulative distribution function F(x) at x = -1 and 1 is 0.5 and 0.5 respectively.
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