By Rachel Quinlan
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Extra info for Introduction to Probability Theory
Example text
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.
85. 8418. 5 Find that real number x for which P(|X| standard normal distribution. 1. We want X > x in 10% of cases. 39 Other Normal Distributions Random variables having normal distributions arise frequently in practice. Of course, not all have expected value 0 and variance 1. 6 A random variable X is said to have the distribution N(µ, σ2 ) if X has pdf fX given by 1 x−µ 2 1 fX (x) = √ e− 2 ( σ ) , 2πσ where µ ∈ R and σ > 0 are fixed. R EMARK: If σ = 1, then fX (x) = f(x − µ), where f is the pdf of the the N(0, 1) distribution.
3. Suppose a, b ∈ R, a < b. Then P(a X b) = P(X b) − P(X = FX (b) − FX (a) b = a f(f)dt − ∞ b = a) f(t)dt −∞ f(t)dt. a b P(a X b) = f(t)dt. 7 Let X be a continuous random variable with pdf f. Then the expectation of X is defined by ∞ E(X) = tf(t)dt. 4, show that E(X) = π. 9 Var(X) = E((X − E(X))2 ): variance of X. This is also given by E(X2 ) − (E(X))2 (this is true also in the discrete case). ∞ Fact: E(X2 ) = t2 f(t)dt. So −∞ ∞ Var(X) = 2 ∞ 2 t f(t)dt − (E(X)) = −∞ ∞ 2 tf(t)dt t f(t)dt − . −∞ −∞ The standard deviation σ of X is defined by σ = 2 Var(X).