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Unformatted text preview: THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT 1301 PROBABILITY AND STATISTICS I
EXAMPLE CLASS 6
Continuous Uniform Distribution
A random variable X has a continuous uniform distribution , , if f for . Exponential Distribution
A random variable X has an exponential distribution if f
Gamma Distribution , A random variable X has a gamma distribution if f for 0, where , for Γ 0. 0, where 0 and 0. Chi-square Distribution
A random variable X has a chi-square distribution of f / , Γ/ for 0. The parameter v is referred as the degrees of freedom.
A random variable X has a normal distribution if f
A random variable X has a beta distribution if f
where Ex.1 0 and 0. , for √
Γ Γ 1 ∞
, ∞, where σ
for 0 0. 1, A random variable Y is said to follow the double exponential distribution if it has the density function
(a) Find the distribution function of Y.
(b) Find the moment generating function of Y.
(c) Find E(Y) and Var(Y). Ex.2 Peggy figures that the total number of thousands of miles that an auto can be driven before it would need
to be junked is an exponential random variable with parameter . Josephine has a used car that she
claims has been driven only 10,000 miles. If Peggy purchases the car, what is the probability that she
would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime
mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly
distributed over (0, 40). Ex.3 The lung cancer hazard rate of a t-year-old male smoker is given by
Assuming that a 40-year-old male smoker survives all other hazards, what is the probability that he
(a) age 50 and
(b) age 60 without contracting lung cancer? Ex.4 The length of a certain kind of item is distributed normally with mean 5.8 cm and standard deviation
0.075 cm. Any item whose length exceeds 6.0cm must be scrapped. Find the proportion of scraps in a
large batch of such items. Ex.5 If X has a Chi-square distribution with degrees of freedom 23, find
(a) Pr 10.2
(b) a and b such that Pr
0.95 and Pr
0.025, where a and b are constants; and
(c) the mean and variance of X. ...
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