Probability Distribution Problems: Solved!

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Hey guys! Let's dive into some probability distribution problems. We'll break down each step so it's super easy to follow. We'll tackle finding the value of 'c' for a probability distribution, calculating the cumulative distribution function, and determining probabilities for specific events. Let's get started!

CPMK 1: Understanding Probability Distributions (30 Points)

Let's start with a classic probability problem. We are given a function and need to figure out some key things about its probability distribution. It’s like detective work with numbers, super cool, right?

Given Information

We have a function defined as:

f(x) = c(x^2 + 4), for x = 0, 1, 2, 3;

This function, guys, is our starting point. It tells us how probability is distributed across different values of x. The constant 'c' is crucial, it's what makes this a probability distribution. We'll figure out its value in the first part.

a) Finding the Value of 'c' for a Valid Probability Distribution

The first question asks us to find the value of c that makes this function a valid probability distribution. Now, what does that even mean? Well, for a function to be a probability distribution, the sum of probabilities for all possible values of x must equal 1. Think of it like this: if you list all possible outcomes, the chance that something happens has to be 100%, right?

So, what we need to do is calculate the probability for each value of x (0, 1, 2, and 3), add them up, and set the total equal to 1. This gives us an equation we can solve for c. Let's break it down step by step:

  1. Calculate f(x) for each x:

    • f(0) = c(0^2 + 4) = 4c
    • f(1) = c(1^2 + 4) = 5c
    • f(2) = c(2^2 + 4) = 8c
    • f(3) = c(3^2 + 4) = 13c

  2. Sum the probabilities:

    4c + 5c + 8c + 13c = 30c

  3. Set the sum equal to 1 and solve for c:

    30c = 1

    c = 1/30

Boom! We found c. So, c = 1/30 is the magic number that makes our function a legitimate probability distribution. This means the probabilities will all add up to 1, like they're supposed to.

b) Determining the Cumulative Distribution Function (CDF)

Next up, we need to find the cumulative distribution function (CDF). This sounds fancy, but it's actually a pretty straightforward concept. The CDF, usually written as F(x), tells us the probability that the random variable X is less than or equal to a certain value x. Think of it as adding up probabilities as you move along the x-axis.

To find the CDF, we need to calculate the cumulative probability for each value of x (0, 1, 2, and 3). Remember, we now know that c = 1/30.

  1. F(0): This is the probability that X is less than or equal to 0. So, F(0) = P(X ≤ 0) = f(0) = (1/30) * (0^2 + 4) = 4/30

  2. F(1): This is the probability that X is less than or equal to 1. So, F(1) = P(X ≤ 1) = f(0) + f(1) = 4/30 + (1/30) * (1^2 + 4) = 4/30 + 5/30 = 9/30

  3. F(2): This is the probability that X is less than or equal to 2. So, F(2) = P(X ≤ 2) = f(0) + f(1) + f(2) = 9/30 + (1/30) * (2^2 + 4) = 9/30 + 8/30 = 17/30

  4. F(3): This is the probability that X is less than or equal to 3. So, F(3) = P(X ≤ 3) = f(0) + f(1) + f(2) + f(3) = 17/30 + (1/30) * (3^2 + 4) = 17/30 + 13/30 = 30/30 = 1

And there you have it! We've calculated the CDF for our probability distribution. Notice that F(3) equals 1, which makes sense because it represents the probability of X being less than or equal to the highest possible value (3), so it's certain to happen.

c) Calculating P(X < 2)

Finally, we need to find the probability that X is strictly less than 2. This is a little different from the CDF, which includes the value 2. To find P(X < 2), we need to add up the probabilities for all values of x that are less than 2, which are 0 and 1.

P(X < 2) = P(X = 0) + P(X = 1) = f(0) + f(1) = 4/30 + 5/30 = 9/30

So, the probability that X is less than 2 is 9/30. We're on a roll!

CPMK2: Let's Talk Distributions (20 Points)

Now, let's shift gears a bit and consider a scenario involving distributions. We'll build on the concepts we've learned to tackle a new problem. This is where things get even more interesting!

Discussion Category: Mathematics

This section looks like it's the start of a new problem, but the provided snippet ends abruptly. We need more information to solve this problem. However, it's a great starting point for a discussion on probability distributions in mathematics. We could explore different types of distributions (like binomial, Poisson, normal), their properties, and how they are used in real-world applications. For example, we could discuss:

  • Binomial Distribution: This distribution models the probability of obtaining a certain number of successes in a fixed number of independent trials, each with the same probability of success. Think of flipping a coin multiple times and counting the number of heads.
  • Poisson Distribution: This distribution models the probability of a certain number of events occurring in a fixed interval of time or space, given that these events occur with a known average rate and independently of the time since the last event. Think of the number of customers arriving at a store in an hour.
  • Normal Distribution: This is a continuous probability distribution that is symmetric and bell-shaped. It's one of the most common distributions in statistics and is used to model many natural phenomena, such as height, weight, and test scores.

We could also dive into the properties of these distributions, such as their mean, variance, and standard deviation. Understanding these properties is crucial for making predictions and drawing conclusions based on the data.

To make this section more complete, we need the full problem statement. But hopefully, this gives you a good starting point for thinking about different types of distributions and their applications.

In conclusion, we've successfully tackled a probability distribution problem, calculated the cumulative distribution function, and determined probabilities for specific events. We've also laid the groundwork for a discussion on different types of probability distributions and their properties. Keep practicing, guys, and you'll become probability masters in no time! Let me know if you have any questions – I'm here to help. Cheers!**