Unraveling The Probability Puzzle: PDF Validation And Variance Calculation

Hey everyone! Today, we're diving headfirst into the world of probability and statistics, specifically focusing on a continuous random variable, denoted as X. We've got a probability density function (PDF) at our disposal, and our mission, should we choose to accept it, is to validate this PDF and then calculate its variance. It sounds a bit technical, but trust me, we'll break it down into bite-sized pieces so that everyone can follow along. This is all about understanding the core concepts of probability distributions, which are absolutely crucial in various fields, including data science, finance, and engineering. Let's get started!

Proving the Validity of the Probability Density Function (PDF)

Alright, first things first, let's make sure our PDF, f(x), is legit. A PDF is like a superhero, and to be a superhero, it needs to meet certain criteria. Specifically, it needs to satisfy two key conditions to be considered valid. These are the backbone of any valid PDF, and we will check them one by one. Our given PDF is defined as:

f(x) = egin{cases} 3x^2, & 0 \le x \le 1 \ 0, & ext{otherwise} First, the function must always be non-negative. This means that for any value of x, the PDF must be greater than or equal to zero. This makes perfect sense if you think about it; the probability of an event can never be negative, right? It's like saying you have negative friends; it doesn't make sense! We're dealing with $3x^2$ when $x$ is between 0 and 1, and it's 0 otherwise. Within the range [0, 1], $x^2$ will always be non-negative, and multiplying by 3 keeps it that way. In areas outside this range, the function is zero, which is also non-negative. Thus, our first condition is met. Second, the integral of the PDF over its entire range must equal 1. This condition ensures that the total probability over all possible outcomes is exactly 1, or 100%. Think of it as the probability of something *definitely* happening. We need to integrate our function from negative infinity to positive infinity and see if we end up with 1. Since our function is only non-zero between 0 and 1, we only need to integrate over that interval: $\int_{-\infty}^{\infty} f(x) dx = \int_{0}^{1} 3x^2 dx

To integrate $3x^2$, we increase the power of x by 1 and divide by the new power (i.e. we use the power rule for integration). So we have:

$$= \left[ x^3 \right]_{0}^{1}$$

Now, we evaluate this at the upper and lower limits:

$$= (1^3) - (0^3) = 1 - 0 = 1$$

And there you have it! The integral of our PDF is indeed equal to 1. This means the second condition is also met.

Therefore, because our f(x) satisfies both of these conditions—non-negativity and the integral equaling 1—we can confidently say that our PDF is valid!

Calculating the Variance of the Random Variable X

Now that we've established the validity of our PDF, let's move on to the fun part: calculating the variance of X, denoted as Var(X). Variance is a measure of how spread out the values of a random variable are. A higher variance means the values are more dispersed, while a lower variance means they're clustered closer together. The variance is a critical concept in statistics, helping us understand the variability or uncertainty associated with a random variable. The calculation involves a few steps, but we'll break them down. We need to use the following formula for variance:

$$\text{Var}(X) = E[X^2] - (E[X])^2$$

Where E[X] is the expected value (or mean) of X, and E[X²] is the expected value of X squared. So, our strategy is to find these expected values first and then plug them into the variance formula.

Step 1: Calculate the Expected Value, E[X] (The Mean)

The expected value (or mean) of a continuous random variable is calculated as:

$$E[X] = \int_{-\infty}^{\infty} x \cdot f(x) dx$$

Given that $f(x) = 3x^2$ for $0 \le x \le 1$ and 0 otherwise, we can rewrite this as:

$$E[X] = \int_{0}^{1} x \cdot 3x^2 dx = \int_{0}^{1} 3x^3 dx$$

Now, let's integrate $3x^3$:

$$E[X] = \left[ \frac{3}{4}x^4 \right]_{0}^{1}$$

Evaluating at the limits:

$$E[X] = \frac{3}{4}(1)^4 - \frac{3}{4}(0)^4 = \frac{3}{4} - 0 = \frac{3}{4}$$

So, E[X] = 3/4. This is the average or mean value of our random variable X.

Step 2: Calculate the Expected Value of X Squared, E[X²]

The expected value of X squared is calculated as:

$$E[X^2] = \int_{-\infty}^{\infty} x^2 \cdot f(x) dx$$

Using our PDF, this becomes:

$$E[X^2] = \int_{0}^{1} x^2 \cdot 3x^2 dx = \int_{0}^{1} 3x^4 dx$$

Now, let's integrate $3x^4$:

$$E[X^2] = \left[ \frac{3}{5}x^5 \right]_{0}^{1}$$

Evaluating at the limits:

$$E[X^2] = \frac{3}{5}(1)^5 - \frac{3}{5}(0)^5 = \frac{3}{5} - 0 = \frac{3}{5}$$

So, E[X²] = 3/5.

Step 3: Calculate the Variance, Var(X)

Now we have everything we need to calculate Var(X). Using the formula:

$$\text{Var}(X) = E[X^2] - (E[X])^2$$

We plug in our calculated values:

$$\text{Var}(X) = \frac{3}{5} - \left(\frac{3}{4}\right)^2 = \frac{3}{5} - \frac{9}{16}$$

To subtract these fractions, we need a common denominator, which is 80:

$$\text{Var}(X) = \frac{48}{80} - \frac{45}{80} = \frac{3}{80}$$

Therefore, the variance of X is 3/80. This tells us how spread out the values of X are around the mean of 3/4.

Wrapping Things Up

And there you have it, folks! We've successfully validated the given PDF and calculated the variance of the random variable X. This process demonstrates a fundamental concept in probability theory and statistics. We've shown how to verify if a function is a valid PDF by checking if the integral equals 1 and that it does not go below zero. Then, we calculated the variance, which tells us about the spread of our data, by finding the expected values and using the appropriate formula. This kind of analysis is very important when it comes to understanding real-world phenomena.

I hope this explanation has been clear and helpful. Remember, understanding PDFs and calculating variance is a core skill in many fields. Keep practicing, and you'll get the hang of it! If you have any more questions or want to delve deeper, don't hesitate to ask. Happy calculating, and keep exploring the fascinating world of probability!