Let's solve the given problem step by step.

Problem Summary

A random variable $X$ follows a probability function as shown in the table:

The tasks are to:

1. Determine the value of $k$.
2. Find $P(X \leq 0)$.
3. Find $P(X < 3)$.
4. Find $P(X > 0)$.
5. Find $F(X = 3)$ in usual notation (the cumulative distribution function up to 3).

Solution

Step 1: Determine the value of $k$

The sum of all probabilities must equal 1: $0.1 + 0.2 + 0.1 + k + 0.1 + 0.2 + 0.1 = 1$

Calculating the sum of the known probabilities: $0.8 + k = 1$

Solving for $k$: $k = 1 - 0.8 = 0.2$

So, $k = 0.2$.

Step 2: Find $P(X \leq 0)$

We need to calculate the probability of $X$ being less than or equal to 0, which includes $X = -2, -1,$ and $0$.

$P(X \leq 0) = P(X = -2) + P(X = -1) + P(X = 0)$ $P(X \leq 0) = 0.1 + 0.2 + 0.1 = 0.4$

So, $P(X \leq 0) = 0.4$.

Step 3: Find $P(X < 3)$

We need the probability of $X$ being less than 3, which includes $X = -2, -1, 0, 1,$ and $2$.

$P(X < 3) = P(X = -2) + P(X = -1) + P(X = 0) + P(X = 1) + P(X = 2)$ $P(X < 3) = 0.1 + 0.2 + 0.1 + 0.2 + 0.1 = 0.7$

So, $P(X < 3) = 0.7$.

Step 4: Find $P(X > 0)$

We need the probability of $X$ being greater than 0, which includes $X = 1, 2, 3,$ and $4$.

$P(X > 0) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$ $P(X > 0) = 0.2 + 0.1 + 0.2 + 0.1 = 0.6$

So, $P(X > 0) = 0.6$.

Step 5: Find $F(X = 3)$

The cumulative distribution function $F(X = 3)$ is the probability that $X \leq 3$.

$F(X = 3) = P(X = -2) + P(X = -1) + P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$ $F(X = 3) = 0.1 + 0.2 + 0.1 + 0.2 + 0.1 + 0.2 = 0.9$

So, $F(X = 3) = 0.9$.

Summary of Answers

1. $k = 0.2$
2. $P(X \leq 0) = 0.4$
3. $P(X < 3) = 0.7$
4. $P(X > 0) = 0.6$
5. $F(X = 3) = 0.9$

Would you like a deeper explanation of any part?

Here are some related questions to explore further:

1. How would we calculate $P(X = 3)$ directly from the table?
2. What would the cumulative distribution function look like for all values of $X$?
3. How is the expectation $E(X)$ calculated for this probability distribution?
4. How can we find the variance of $X$ given this probability distribution?
5. How would the probabilities change if the distribution were normalized differently?

Tip: Always ensure that the sum of all probabilities in a distribution is 1; this is a fundamental property of probability distributions.