Continuous Probability Distribution

Photo by Markus Winkler on Unsplash

In my last two articles, we have looked at discrete probability distribution where data consists of distinct numeric values and we can specify the exact values of probability of each of the data. But in real life, maximum problems don’t involve this type of discrete data. In the case of continuous random variables, they can take an infinite number of values with the specific interval. For example, the values between [4,7]. Here the variable can an infinite number of values between 4 and 7. Again we can take the example of the amount of rainfall in a year for a specific city.

Probability Density Function :

When we deal with a discrete probability distribution, we use to produce distribution by showing the probability of each value in a table. But in case of continuous data, it is not possible to assign a probability to each value. Instead, we need to calculate a specific level of accuracy and probability of getting a range of values.

We can describe a continuous probability distribution using the probability density function (pdf).

The probability density function, f(x) is a function that is used to describe probabilities of a continuous random variable across a range of values.

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In the above picture, the function f(x) or probability is constant up to x=10 and then becomes 0 for all other values. Here f(x) represents the height of the curve at point x. For continuous random variables, probabilities are nothing but the area under the curve.

For any continuous probability distribution :

• f(x) ≥ 0 for all values of x.
• The area under the entire curve is equal to 1.

The above curve represents a rectangle with base 10. Since the total area under the curve will become 1, we can write as :

1 = 10 x height (f(x))

f(x) = 1/10 = 0.1

So f(x) must be equal to 0.1 to ensure total area under it will be 1.

f(x) = 0.1 for x is between 0 and 10.

For continuous probabilities, we have to find the probabilities by calculating the area under the probability density curve. To find the area we will use integration.

Suppose, for a random variable X :

f(x) = ax³ for 2≤x≤4 and 0 otherwise.

So, f(x)= 1/60 x³ for 2≤x≤4 and 0 otherwise.

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Suppose, we need to find P(X≥3) for the above function, f(x). Then we have to find the following area of the function.

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We can also calculate the median of the above function. The median will divide the entire area into two halves. Let m be the value of x, which divides the entire area into two halves.

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So, the median for the above function will be 3.425.

Mean and Variance of continuous Probability Distribution :

The mean of a continuous probability distribution function, f(x) is given by :

So, to find the mean of the above function, we can write as :

Again, the variance of a continuous probability distribution function, f(x) is given by :

But, we can also describe variance of X as :

Var(X) = E(X²)- [E(X)]²

We need to find the variance of the above function, f(x) = 1/60 x³, for 2≤x≤4. We will use the above formula of variance. Here, E(X) = 3.3. We have to find E(X²).

So, Var(X) = E(X²)- [E(X)]² = 11.2- (3.3)² = 0.31

That’s all for today. Hope you guys like this article. Keep smiling. 😃