Problems for the Basic Course in Probability (Fall 00) Discrete Probability. Die A has 4 red and white faces, whereas die B has red and 4 white faces. A fair coin is flipped once. If it lands on heads, then die A is rolled; if it lands on tails, then die B is rolled. Show that the probability of getting a red face is /.. A communication system consists of n compartments, each of which functions independently with probability p. The total system will be able to operate effectively if at least one-half of its components function. (a) What is the probability that a 5-component system will be able to operate effectively? (b) For what values of p is a 5-component system more likely to operate effectively than a 3-component system? 3. You have n urns, the rth of which contains r red balls and n r blue balls, r =,..., n. You pick an urn at random and remove balls from it without replacement. Find the probability that the two balls are of different colours. Find the same probability when you put back a removed ball. 4. Per och Pål har elva frukter av vilka tre är giftiga. Per äter 4 på måfå valda frukter och Pål 6; hunden får den återstående. Beräkna sannolikheten att hunden klarar sig. 5. Låt X k, k =,,..., vara oberoende stokastiska variabler med E(X k ) = m och V(X k ) = σ, samt definiera, för n, T n = n k= X k. Beräkna Cov(T n, T n+ ). 6. Tre identiska välbalanserade mynt singlas samtidigt tills alla hamnar med samma sida upp. Hur stor är sannolikheten att mynten singlas mer n n gånger? 7. n balls are placed at random into n cells. Find the probability that exactly two cells remain empty. 8. I en låda ligger tre mynt. Det första myntet har krona på båda sidorna, det andra myntet är symmetriskt med krona på ena sidan och klave på den andra medan det tredje myntet är skevt så att det ger krona i 75% av kasten. Ett av de tre mynten väljs slumpmässigt och kastas. Det kastade myntet visar krona, vad är sannolikheten att det var myntet med krona på båda sidorna som valdes?
9. 0 gifta par går på fest och placeras helt slumpmssigt bredvid varandra vid ett runt bord. Tv eller flera personer av samma kn kan alltså sitta bredvid varandra. Vad är väntevärdet av antalet fruar som har sin man bredvid sig? 0. A fair coin is tossed n times and the number of heads, say N, is counted. Then the coin is tossed N more times. Find the expected number of the heads generated by this process.. Let N be a (random) number of successes in m independent trials with probability r of success. Suppose that conditionally on N = n a variable X has also a binomial distribution with n independent trials and probability p of success. Find the unconditional distribution of X. (Hint: You should get a binomial distribution in the answer).
Continuous Distributions Here we consider distribution functions in form where f(y) > 0, y R, and F (x) = F (+ ) = f ξ (y)dy, f(y)dy =. We say that a random variable ξ has a distribution F (x), if We write F ξ (x) := P {ξ x} = F (x). F ξ (x) = f ξ (y)dy, and call f ξ (y), y R, the density function of ξ. In this case we define the mathematical expectation of ξ by Eξ = xf ξ (x)dy. Also, we can compute for any function g : R R Eg(ξ) = g(x)f ξ (x)dx, () if the integral on the right is well defined. All the properties of the expectations which we proved for the discrete case, hold here as well. In -dim case, let where f(x, y) > 0 and F (x, y) = F (+, + ) = y f(x, y )dy dx, f(y, x)dydx =. Then we define a random vector (ξ, η) by a distribution function F ξ,η (x, y) = P {ξ x, η y} = F (x, y). We write F ξ,η (x, y) = y f ξ,η (x, y )dy dx, 3
and call f ξ,η (x, y) the (joint) density function of (ξ, η). Similar to () we have also for g : R R Eg(ξ, η) = if the integral on the right is well defined. Definition We say that ξ and η are independent if g(x, y)f ξ,η (x, y)dydx, () F ξ,η (x, y) = F ξ (x)f η (y) for all real x and y. ( When the density exists, this happens if and only if for all real x and y.) f ξ,η (x, y) = f ξ (x)f η (y) Exercise. Using formula () derive, that when ξ and η are independent with finite expectations then E(ξη) = EξEη. Examples.. Uniform distribution in [a, b]. We say that X is uniformly distributed over [a, b], if its density is constant within this interval: f X (x) = {, b a if x [a, b], 0, otherwise.. Normal distribution with parameters m, σ. Notation: N(m, σ ). We say that X N(m, σ ), normally distributed, if f X (x) = x e σ, x R. πσ Central Limit Theorem Assume, ξ,..., ξ n,..., are independent identically distributed random variables with Eξ i = m, Vξ i = σ. Then for all x R { ni= ξ i nm P nσ as n. x } π e y dy 4
. Let a basketball player can score on a single shot with probability 0.3. Use the central limit theorem to approximate the probability that out of 5 independent shots he will get at most 5 successful shots.. De stokastiska variablerna X och Y har den simultana täthetsfunktionen Beräkna V (X). f X,Y (x, y) = x e x, x > 0, x < y < x. 3. Let U have a uniform distribution on [ ; ]. Find the density function of U. 5