Rivalry2

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Original 2026-01-01

Modified 2026-02-28

1 To understand the probability distributions and conduct statistical analysis, we need to comprehend the various properties of a continuous random variable. Several important properties distinguish the continuous random variable from the<a>discrete random variable</a>. They include:

2 Probability Density Function

3 A probability density function f(x) defines a continuous random variable x. The relative probability for x to be a certain value x is represented by the PDF f(x). The PDF needs to meet two conditions. The first one is f(x) ≥ 0 for all x. It means the probability density function (PDF) must always be non-negative but can be zero for some values. The next condition is that the total probability over all possible values of x must be 1, meaning the<a></a><a>sum</a>of all probability values must be 100% or 1. This is written as:

4 \(\int_{-\infty}^{\infty} f(x)\, dx = 1 \)

5 Cumulative Distribution Function (CDF)

6 The cumulative distribution function (CDF), represented as F(x), shows the likelihood that a continuous random variable X will take a value that is less than or equal to x:

7 \(F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t)\,dt \)

8 The CDF is non-decreasing and continuous. The CDF approaches 0 as x moves closer to negative infinity.

9 lim F(x) = 0 x→∞

10 Likewise, the CDF approaches 1 as x moves closer to positive infinity.

11 lim F(x) = 1 x→∞

12 Moment Generating Function (MGF)

13 The moment generating function (MGF) of a continuous random variable X is represented as Mx (t). It is defined as:

14 \(M_X(t) = E\!\left(e^{tX}\right) = \int_{-\infty}^{\infty} e^{tx} f(x)\,dx \)

15 The MGF can be used to find all the moments of X, including its mean and <a>variance</a> if it exists.

16 Characteristic Function

17 The characteristic function of a continuous random variable X is expressed as ϕX(t). It is the Fourier transform of the probability density function (PDF).

18 \(\varphi_X(t) = E\!\left(e^{itX}\right) = \int_{-\infty}^{\infty} e^{itx} f(x)\,dx\)

19 It determines the distribution of X uniquely.

20 Mean and Variance of Continuous Random Variable

21 A continuous random variable X with PDF f(x) has the following expectation or mean:

22 \(E(X) = \int_{-\infty}^{\infty} x f(x)\,dx \)

23 The formula represents the expected value of X.

24 The variance of X measures the<a>average</a>of the squared deviations of the random variable from the mean. It is defined as:

25 \(\operatorname{Var}(X) = E\!\left[(X - E(X))^{2}\right] = \int_{-\infty}^{\infty} (x - \mu)^{2} f(x)\,dx \)

26 Here, μ = E (X) is the mean.

27 Where X's second moment is represented by E(X2) and it is denoted as:

28 \(E(X^2) = \int_{-\infty}^{\infty} x^2 f(x)\,dx \)