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Roll a fair die repeatedly. Let X be the number of 6’s in the first 10 rolls and let Y the number of rolls needed to obtain a 3.
(a) Write down the probability mass function of X.
(b) Write down the probability mass function of Y .
(c) Find an expression for P(X ≥ 6).
(d) Find an expression for P(Y > 10).
Given the Standard Normal distribution, find the following
(a) P(Z < 1.8)
(b) P(−1.1 < Z ≤ 1.8)
(c) P(−1.8 ≤ Z ≤−1.1)
(d) P(Z > −2.5)
(e) P(Z > −0.95)
(f) P(Z < −0.95)
(g) P(Z ≥ 2.18)
(h) P(Z > 10)
The number of claims per month paid by an insurance company is modelled by a random variable N with p.m.f satisfying the relation
p(n + 1) =
1/3p(n), n = 0,1,2,...
where p(n) is the probability that n claims are filed during a given month
(a) Find p(0).
(b) Calculate the probability of at least one claim during a particular month given that there have been at most four claims during the month.
The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem?
Probability distribution for a random variable x which corresponds to the number of pens that you have in your bag
(a) The random variable Y has a Poisson distribution and is such that P(Y = 0) = P(Y = 1). What is P(Y 2 = 1)?
(b) Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?
Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that
(a) no more than three customers arrive?
(b) at least two customers arrive?
(c) exactly four customers arrive?
A salesperson has found that the probability of a sale on a single contact is approximately .03. If the salesperson contacts 100 prospects, what is the approximate probability of making at least one sale?
Let Y have a Poisson distribution with mean λ . Find E [Y (Y −1)] and then use this to show that Var(Y ) = λ
A corporation is sampling without replacement for n = 3 firms to determine the one from which to purchase certain supplies. The sample is to be selected from a pool of six firms, of which four are local and two are not local. Let Y denote the number of non-local firms among the three selected. Find:
(a) P(Y = 1)
(b) P(Y ≥ 1)
(c) P(Y ≤ 1)
#331389 on Nov 2022