Introduction to Probability Theory by Rachel Quinlan

By Rachel Quinlan

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X→−∞ The graph of f looks like : R EMARK: It is possible using polar coordinates to show that ∞ f(x)dx = 1. 1 A continuous random variable having the above f as a pdf is said to have the standard normal distribution. Such a random variable is said to have distribution N(0, 1). 2 If X has distribution N(0, 1), show that E(X) = 0 and Var(X) = 1. 37 Outline of Solution: ∞ 1 1 2 √ e− 2 t dt. 2π −∞ Note that the integrand here is an odd function, hence the integral is zero (provided that the improper integral converges which is not difficult to check).

85m. 9 Assume that the flight time (from takeoff to landing) from Dublin Airport to London Heathrow is normally distributed with a mean of 50 minutes and a standard deviation of 5 minutes. (a) What is the probability that the flight time will exceed one hour? (b) What is the probability that the flight time will be between 45 and 55 minutes? (c) Below what duration can we expect the fastest 10% of flights? e. 6826. e. we expect 68% of observed values to be within one standard deviation of the mean.

Var(X) = t √ e− 2 t dt − (E(X))2 = √ 2π 2π −∞ −t2 Integrating by parts with u = t, v = te gives 2 1 1 2 Var(X) = − √ te− 2 t 2π ∞ 1 +√ 2π −∞ ∞ 1 2 t2 e− 2 t dt. −∞ 1 2 e− 2 t dt = 0 + 1 = 1. 3 Suppose that X is a random variable having the distribution N(0, 1). 0) Solution: Let Φ : R −→ [0, 1] be the cdf of X. (The symbol Φ is the upper case of the Greek letter phi). 0) = √ 2π 2 t2 e− 2 dt. −∞ t2 P ROBLEM: The expression e− 2 does not have an antiderivative expressible in terms of elementary functions.

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