CDF 概率计算

Reconstructing the CDF

I'm carefully plotting the cumulative distribution function (CDF) F(x) based on the user’s description, identifying jumps and constant segments, and ensuring continuity and piecewise definition.

Mapping out behavior

OK, let me see: For $x < -1$, $F(x) = 0$. At $x = -1$, $F(x)$ jumps to 0.2, and it stays 0.2 for $-1 < x < 0$.

Analyzing function dynamics

It's interesting to see that from $x = -1$ to $x = 0.2$, $F(x)$ stays steady at 0.2. Then from $x = 1$ to $x = 3$, it increases linearly to 0.6 and stays constant afterward until $x = 5$, where it jumps to 1.

Mapping out the CDF

The CDF $F(x)$ is defined piecewise:

• $F(x) = 0$ for $x < -1$,
• $0.2$ for $-1 \leq x \leq 1$,
• $0.2 + 0.2(x - 1)$ for $1 < x < 3$,
• $0.6$ for $x = 3$,
• $0.6$ for $3 < x \leq 4$ (constant),
• $0.6 + 0.4(x - 4)$ for $4 < x \leq 5$,
• $1$ for $x \geq 5$.

Mapping the increase

I’m thinking of F(x) as a piecewise function, increasing linearly from 0.6 to 1 between $x=4$ and $x=5$, with different expressions for other ranges.

Determining discrete probabilities

I'm exploring discrete probabilities using $P(X = x) = F(x) - \lim_{t \to x^-} F(t)$. Notably, at $x = -1$, there's a jump from $F(-1^-) = 0$ to $F(-1) = 0.2$, indicating a discrete probability scenario.

Assessing probabilities
I’m determining $\Pr(X = -1) = 0.2$ by examining the jump size at $x = -1$. For $\Pr(X < 0)$ and $\Pr(X \leq 0)$, I evaluate the cumulative distribution function. Finally, I’m checking for the jump at $x = 1$.

Breaking down probabilities

I’m looking at the probability calculations for different ranges. For $0 < X \leq 3$, I subtract $F(0)$ from $F(3)$ to get 0.4. For $0 < X < 3$, I just use $F(3^-) - F(0)$ since there’s no jump at $x = 3$.

Confirming the probability

Next, I observed that since $F(0^-)$ equals $F(0)$ at 0, and there is no jump at $x=0$, $\Pr(0 \leq X \leq 3)$ consistently results in 0.4, which aligns with previous calculations.

Evaluating probabilities

I’m calculating specific probabilities given the cumulative distribution function $F(x)$. For $\Pr(0 \leq X \leq 3)$, it’s $F(3) - F(0^-) = 0.4$. For $\Pr(1 < X \leq 2)$, I’m determining $F(2) - F(1)$ using the linear trend in the distribution.

Confirming the calculations

I calculated $F(2)$ and $F(1)$ to determine $\Pr(1 < X \leq 2)$, confirming no jump at $x=1$.

Assessing probabilities

I’m calculating $\Pr(1 \leq X \leq 2)$ and $\Pr(X > 5)$ using cumulative distribution function (CDF) values. Interestingly enough, there’s no jump at $x=1$ and $x=5$.

Calculating probabilities

I’m curious about the probability for different ranges of $X$. For $\Pr(X \geq 5)$, since $F(5^-) = 1$ and there's no jump at $x=5$, the probability is $0$. For $\Pr(3 \leq X \leq 4)$, $F(x)$ is constant at $0.6$ from $x=3$ to $x=4$, so the probability remains $0.6$.