properties of expectation

Let X 1 and X 2 be two random variables and c 1;c 2 be two real numbers, then E[c 1X 1 + c 2X 2] = c 1EX 1 + c 2EX 2: Taking these two properties, we say that expectation is a positive linear functional. Each new coupon obtained is type $i$ with probability $p_{i},$ where $p_{1}=p_{2}=1 / 8, p_{3}=$ $p_{4}=3 / 8 .$ Find the expected number of coupons that one must obtain to have at least one of,If $X$ and $Y$ are independent and identically distributed with mean $\mu$ and variance $\sigma^{2},$ find.In Problem $6,$ calculate the variance of the sum of the rolls.In Problem $9,$ compute the variance of the number of empty urns.If $E[X]=1$ and $\operatorname{Var}(X)=5,$ find,If 10 married couples are randomly seated at a round table, compute (a) the expected number and,Cards from an ordinary deck are turned face up one at a time. Find,There are $n+1$ participants in a game.

A player throws a fair die and simultaneously flips a fair coin. If X(s) 0 for every s2S, then EX 0 2.

Let $p_{i}$ denote the proportion of the population that is in subgroup $i, i=1, \ldots, r .$ If the average weight of the members of subgroup $i$ is $w_{i}, i=1, \ldots, r$ what is the average weight of the members of the population?A prisoner is trapped in a cell containing 3 doors. Let and be constants. The third door leads to freedom after 1 day of travel. P.IVA 06333200829 REA PA-314445.

Let $A$ denote a specificd one of the players, and let $X$ denote the amount that is received by $A$(a) Compute the expected total prize shared by the players.
Consequently, (b) Law of total expectation.

The first door leads to a tunnel that returns him to his cell after 2 days' travel.
At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1-p$ is put in its place. (For instance, if 4 people win, then cach of them receives $\frac{1}{4},$ whereas if there are no winners, then none of the participants receive anything.)

Each insect caught will, independently of the types of the previous catches, be of type $i$ with probability,In an urn containing $n$ balls, the $i$ th ball has weight $W(i), i=1, \ldots, n .$ The balls are removed without replacement, one at a time, according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn.

Properties of conditional expectation (a) Linearity. If the discretized regions are determined by $a_{0}=0, a_{1}=\frac{1}{2},$ and $a_{2}=1$ calculate the optimal quantizer $Y$ and compute $E\left[(X-Y)^{2}\right]$,The moment generating function of $X$ is given by $M_{X}(t)=\exp \left\{2 e^{t}-2\right\}$ and that of $Y$ by $M_{Y}(t)=$,Let $X$ be the value of the first die and $Y$ the sum of the values when two dice are rolled. If the sum is not $7,$ then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For instance, if $n=$.A group of $n$ men and $n$ women is lined up at random.A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Upon arrival, each person looks to see if he or she has any friends among those present. Hint: Let $X_{i}$ denote the return when you use the critical value $i .$ To compute $E\left[X_{i}\right],$ condition on the initial sum.Ten hunters are waiting for ducks to fly by. Each person independently is a winner with probability $p .$ The winners share a total prize of 1 unit.

In Example $5 \mathrm{i}$ we showed that, for $0 \leq x \leq 1, E[N(x)]=e^{x},$ where,An urn contains 30 balls, of which 10 are red and 8 are blue. 4 or a bad year with probability. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body's mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. If $E[X]=1.8$ what is (a) the largest possible value of $P\{X=3\} ?$,Consider $n$ independent flips of a coin having probability $p$ of landing on heads. Compute the joint moment generating function of $X$ and $Y$.Two envelopes, each containing a check, are placed in front of you. If the sum is $7,$ then the game ends and you win 0.

Find $\operatorname{Cov}(X, Y)$,Type $i$ light bulbs function for a random amount of time having mean $\mu_{i}$ and standard deviation $\sigma_{i}, i=1,2 .$ A light bulb randomly chosen from a bin of bulbs is a type 1 bulb with probability $p$ and a type 2 bulb with probability $1-p .$ Let $X$ denote the lifetime of this bulb.

One of each is randomly chosen and the object of the game is to guess the chosen three. Is it possible to devise a strategy that does better than just accepting the first envelope?Successive weekly sales, in units of one thousand dollars, have a bivariate normal distribution with common mean $40,$ common standard deviation 6 and correlation .6. If a small pill is chosen, then that pill is eaten.

Determine her expected winnings.

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