The lifespan of an electrical component has an exponential
distribution with parameter lambda = 0.013. Suppose we have an iid
sample of size 100 of these components
Some hints: P(X < c) for an exponential(lambda) can be found via
pexp(c,lambda) E[X] = 1/lambda and Var[X] = 1/lambda^2
Round all answers to 4 decimals
Using the exact probability distribution, what is the probability
that a single component will be within 15.38 units of the
population mean?
Using Chebyshev's inequality, what is...
2. Let X be exponential with rate lambda. What is the pdf of Y =
X^0.5? How about Y = X^3? Contrast the complexity of this result to
transformation of a discrete random variable.
Let X and Y be independent and identical uniform distribution on
[0, 1]. Let Z=min(X, Y). Find E[Y-Z].
Hint: condition on whether Y=Z or not. What is the probability
Y=Z?
1. Let X be the uniform distribution on [-1, 1] and let Y be the
uniform distribution on [-2,2].
a) what are the p.d.f.s of X and Y resp.?
b) compute the means of X, Y. Can you use symmetry?
c) compute the variance. Which variance is higher?
Assume X is a Poisson random variable with parameter lambda.
What is the test statistic and the best critical region for
testing
H_0:lambda = lambda_0
versus H_a:lambda = lambda_1
where lambda_0 < lambda_1 are constants?
Let X and Y have joint discrete distribution p(x, y) = 3 20 (.5
x ) (.7 y ), x = 0, 1, 2, . . . , and y = 0, 1, 2, . . .. Find the
marginal probability function P(X = x). [hint: for a geometric
series X∞ n=0 arn with −1 < r < 1, r 6= 0, then X∞ n=0 arn =
a 1 − r ]
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
Let X and Y be i.i.d. geometric random variables with parameter
(probability of success) p, 0 < p < 1. (a) (6pts) Find P(X
> Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y = n), for
n = 2, 3, ..., and k = 1, 2, ..., n − 1