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ST2334: SOME NOTES ON CONTINUOUS RANDOM VARIABLES In the following note we give some notes on continuous random variables. Consider a discrete random variable X with support X and PMF f (x). Then we have learnt that P(X = x) = f (x) x ∈ X. Now for a continuous random variable Y with support Y (lets say that Y has ‘nothing to do’ with X, for example associated to a different experiment) and PDF g(y). Then P(Y = y) = 0 y ∈ Y that is, in general, one has P(Y = y)is not normally equal to g(y). For X (the discrete random variable) we had learnt that the distribution function F (x) is X F (x) = f (u). {u≤x∩u∈X} Similarly, for Y we have for any y ∈ Y Z G(y) = g(u)du. {u≤y∩u∈Y} If one thinks as g(y) as a continuous function, the distribution function represents the area under the curve, to the left of y. More generally we had for X: X P(X ∈ A) = f (x). x∈A Now for our continuous random variable Z P(Y ∈ B) = g(y)dy. B In essence, all we are doing is replacing summation with integration. Note that all the rules we have are repeated; e.g. for B1 and B2 two disjoint sets Z Z Z P(Y ∈ B1 ∪B2 ) = g(y)dy = g(y)dy+ g(y)dy = P(Y ∈ B1 )+P(Y ∈ B2 ). B1 ∪B2 B1 B2 We note also, if Y = [a, b], −∞ < a < b < ∞ G(a) = 0 G(b) = 1. If instead Y = [0, ∞): G(0) = 0 lim G(u) = 1. u→∞ Finally if Y = (−∞, ∞) lim G(u) = 0 lim G(u) = 1 u→−∞ u→∞ 1 2 ST2334 Example Suppose that X ∼ N (µ, σ 2 ), that is the PDF of X is o n 1 1 x ∈ X = R. f (x) = √ exp − 2 (x − µ)2 2σ σ 2π Let us find the distribution function. First suppose that µ = 0 and σ 2 = 1; in this situation, we call X a standard normal random variable. Now define Z x n 1 o 1 √ exp − u2 du. P(X ≤ x) = Φ(x) = 2 2π −∞ In general, the integral on the RHS cannot be computed in a closed form, but by now there are very accurate approximations on a computer. Second, let us consider the general case. We have Z x o n 1 1 √ exp − 2 (u − µ)2 du P(X ≤ x) = 2σ −∞ σ 2π Z x−µ n 1 o σ 1 √ exp − v 2 dv = 2 2π −∞ x − µ = Φ . σ We made the substitution v = (u − µ)/σ to go to the second line. This substituion is called standardization for normal random variables. Given a computer program to calculate Φ(x) we can then compute a variety of probabilities associated to normal random variables. For example, if we want to calculate the probability that X ∈ [a, b] for some −∞ < a < b < ∞, we have P(X ∈ [a, b]) = P(X ≤ b) − P(X ≤ a) a − µ b − µ −Φ . = Φ σ σ To obtain the first equality, we remark that P(X ∈ (−∞, b]) = P(X ∈ (−∞, a) ∪ [a, b]) = P(X ∈ (−∞, a)) + P(X ∈ [a, b]) so we have P(X ∈ (−∞, b]) = P(X ∈ (−∞, a))+P(X ∈ [a, b]) ⇒ P(X ∈ [a, b]) = P(X ∈ (−∞, b])−P(X ∈ (−∞, a)). We then note P(X ∈ (−∞, b]) = P(X ≤ b) and P(X ∈ (−∞, a)) = P(X ∈ (−∞, a]) (note that P(X = a) = 0). If this explanation is confusing; we are just saying the area under the curve between [a, b] (for a continuous function) is the difference between all the area to the left of b take away all the area to the left of a. Similarly for −∞ < a < b < c < d < ∞: b − µ a − µ d − µ c − µ −Φ +Φ −Φ . P(X ∈ [a, b]∪[c, d]) = P(X ∈ [a, b])+P(X ∈ [c, d]) = Φ σ σ σ σ