Topics
Continuous random variables
0 of 0 exercises completed
Continuous random variables with pdfs, probabilities as areas under \(f(x)\), normalization to total area 1, and finding \(E(X)\), median, mode, variance, standard deviation, and the effects of linear transformations \(aX+b\).
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
Concept of Probability density
AHL 4.14
A continuous random variable X is a random variable that can take any value in a given interval. Since there are infinitely many possible values within any interval (finite or infinite), the probability of any specific value is 0. Instead, you have to consider the probability that the value of X will fall within some specific range.
This probability is the area under a curve:
P(a≤X≤b)=∫abf(x)dx🚫
The function f(x) is called the probability density function. Its values are not probabilities - since P(X=x)=0 - but instead an abstract measure of how "densely packed" the probability is around each point.
Normalization
AHL 4.14
Since the probability of any sample space must be equal to 1,
P(−∞≤X≤∞)=∫−∞∞f(x)dx=1🚫
If a probability density function is only defined for an interval a≤X≤b, then it is zero everywhere else and the following integrals are equivalent:
∫−∞∞f(x)dx=∫abf(x)dx=1
For a probability density function g(x) where ∫−∞∞g(x)=1, finding the value of k such that ∫−∞∞kg(x)=1 is called normalizing the probability function.
Expected Value of a Continuous Random Variable
AHL 4.14
The expected value of a continuous random variable is
E(X)=μ=∫−∞∞xf(x)dx📖
Intuitively, this can be thought of as the "balance point" of the distribution, the weighted sum of all values of x.
The expected value is also known as the mean.
Variance & SD a Continuous Random Variable
AHL 4.14
The variance of a continuous random variable X is given by
Var(X) =E[(X−μ)2]=E(X2)−[E(X)]2📖 =∫−∞∞x2f(x)dx−μ2📖
The standard deviation can be found by taking the square root of this result.
Linear Transformation of Random Variables
AHL 4.14
Suppose that the random variable X is scaled and shifted, producing the random variable aX+b. The expected value and variance of the resulting variable are
E(aX+b)Var(aX+b)=aE(X)+b=a2Var(X)
Median of a Continuous Random Variable
AHL 4.14
The median of a continuous random variable X is the value m that splits the distribution into two equal areas:
∫−∞mf(x)dx=∫m∞f(x)dx=21🚫
Mode of a Continuous Random Variable
AHL 4.14
The mode of a continuous random variable X is that value x that maximizes f(x). On a graph, this corresponds to the peak of the probability density function.
Nice work completing Continuous random variables, here's a quick recap of what we covered:
Skills covered
Exercises checked off
I'm Plex, here to help you understand this concept!