Consider a random variable Xthat takes values +1 and -1 with probabili...
The p.d.t of the random variable is
The cumulative distribution function F(x) is the probability uptox as given below:
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Consider a random variable Xthat takes values +1 and -1 with probabili...
Cumulative Distribution Function (CDF):
The cumulative distribution function (CDF) of a random variable X is a function that gives the probability that X will take on a value less than or equal to x. It is denoted by F(x).
Given Information:
In this problem, we are given a random variable X that takes values 1 and -1 with equal probabilities of 0.5 each. We need to find the values of the cumulative distribution function (CDF) at x = -1 and x = 1.
Solution:
To find the CDF, we need to calculate the cumulative probabilities for each value of X.
Cumulative Probability for X = -1:
The cumulative probability for X = -1 is the sum of the probabilities of all values of X less than or equal to -1.
Since X can only take on the values 1 and -1, the cumulative probability for X = -1 is equal to the probability of X = -1.
P(X ≤ -1) = P(X = -1) = 0.5
Hence, the value of the CDF at x = -1 is 0.5.
Cumulative Probability for X = 1:
Similarly, the cumulative probability for X = 1 is the sum of the probabilities of all values of X less than or equal to 1.
Since X can only take on the values 1 and -1, the cumulative probability for X = 1 is equal to the sum of the probabilities of X = -1 and X = 1.
P(X ≤ 1) = P(X = -1) + P(X = 1) = 0.5 + 0.5 = 1
Hence, the value of the CDF at x = 1 is 1.
Summary:
The values of the cumulative distribution function (CDF) at x = -1 and x = 1 are 0.5 and 1, respectively.
Therefore, the correct answer is option C: 0.5 and 1.