Let \, X\, be a random varible with invertible cdf \, F_ X. Define another random variable \, Y=F_ X(X). Find the cdf F_ Y of \, Y\,.
For t<0:
F_ Y(t)=\quad
unanswered
For t\geq 1:
F_ Y(t)=\quad
unanswered
For \, 0\leq t< 1:
F_ Y(t)=\quad
To find the cdf of Y (denoted as FY), we need to evaluate it for different values of t.
For t < 0, Y = FX(X) takes on a value less than 0 with probability 0 (since FX only takes values between 0 and 1). Therefore, FY(t) = 0.
For t >= 1, Y = FX(X) takes on a value greater than or equal to 1 with probability 1 (since FX only takes values between 0 and 1 and maps to [0, 1]). Therefore, FY(t) = 1.
For 0 <= t < 1, we need to find the probability that Y = FX(X) is less than or equal to t. Since FX(X) is the inverse of FX, this is equivalent to finding the probability that X is less than or equal to FX^(-1)(t). We can express this as:
FY(t) = P(Y <= t) = P(FX(X) <= t) = P(X <= FX^(-1)(t)) = FX(FX^(-1)(t)) = t
Therefore, for 0 <= t < 1, FY(t) = t.
To find the cumulative distribution function (CDF) of the random variable Y, we need to consider the values of t in different intervals.
1. For t < 0:
Since Y is defined as F_X(X), which is the probability that X is less than or equal to X, the CDF of Y for t less than 0 is 0. Therefore:
F_Y(t) = 0, for t < 0.
2. For t >= 1:
Similarly, if t is greater than or equal to 1, it means that Y takes on the maximum value of 1, since F_X(X) is restricted between 0 and 1. Therefore:
F_Y(t) = 1, for t >= 1.
3. For 0 <= t < 1:
In this interval, we can use the properties of the CDF to find F_Y(t). Let's break it down into steps.
Step 1: Calculate the probability that X is less than or equal to x, i.e., P(X <= x), for a given x. This is equal to F_X(x).
Step 2: Invert the CDF to find the value of x such that F_X(x) is equal to t, i.e., x = F_X^(-1)(t).
Step 3: Calculate the probability that F_X(X) is less than or equal to t, i.e., P(F_X(X) <= t). Since Y is defined as F_X(X), this probability is equivalent to P(Y <= t).
Therefore, for 0 <= t < 1:
F_Y(t) = P(Y <= t) = P(F_X(X) <= t) = P(X <= F_X^(-1)(t)) = F_X(F_X^(-1)(t)) = t.
In other words, for 0 <= t < 1:
F_Y(t) = t.