Probability density, normalization, expectation and variance of c.r.v.'s. median and mode of c.r.v's
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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:
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.
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Since the probability of any sample space must be equal to 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:
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.
The expected value of a continuous random variable is
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.
The variance of a continuous random variable X is given by
The standard deviation can be found by taking the square root of this result.
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
The median of a continuous random variable X is the value m that splits the distribution into two equal areas:
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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.
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