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What is the formula for a random variable?
The formula, if any, depends on the probability distribution function for the variable. In the case of a discrete variable, X, this defines the probability that X = x. For a continuous variable, the probability density function is a continuous function, f(x), such that Pr(a < X < b) is the area under the function f, between a and b (or the definite integral or f, with respect to x, between a and b.
What is exponential distribution?
The exponential distribution is a continuous probability distribution with probability density definded by: f(x) = ke-kx for x ≥ 0 and f(x) = 0 otherwise.
Which distribution emits a probability density function f x equals 1 over square root of 2 pi times e to the power of minus x squared divided by 2?
That's a Gaussian distribution.
How do you make predictions using theoretical probability?
The answer depends on what you are trying to predict. Suppose you have a discrete random variable X with a probability density function p(X) = prob(X = x), then the expected value of a function f(X) of X is the sum of f(x)*p(x), summed over all possible values of x. For a continuous variable, the procedure is similar, except that you need to integrate rather than sum.
What is a cumulative distributive function?
It the the probability that the random variable in question takes any value up to and including the argument. Suppose you have a random variable X and f(x) is the probability that X = x [that is, the rv X takes the value x]. If F(x) denotes the cumulative distribution function of X, then F(x) is the sum of all f(y) where y <= x. Thus, for a fair die, F(1) = f(1) = 1/6 F(2) = f(1) + f(2) = 2/6 F(3) = f(1) + f(2) + f(3) = 3/6 and so on. Note that F(X) = 0 for X < 1, F(a+b) where a is an integer in the interval [1,6] and 0<b<1 is F(a). Thus, for example, F(3.5) = F(3). and F(x) = 1 for x >=6. In the case of continuous probability distributions, the summation is replaced by integration.
