11.2 Problems on mathematical expectation  (Page 2/6)

(See Exercise 19 from "Problems on Distribution and Density Functions"). The number of noise pulses arriving on a power circuit in an hour is a randomquantity having Poisson (7) distribution. What is the expected number of pulses in an hour?

(See Exercise 24 and Exercise 25 from "Problems on Distribution and Density Functions"). The total operating time for the units in Exercise 24 is a random variable $T\sim$ gamma (20, 0.0002). What is the expected operating time?

(See Exercise 41 from "Problems on Distribution and Density Functions"). Random variable X has density function

What is the expected value $E\left[X\right]$ ?

Truncated exponential. Suppose $X\sim$ exponential $\left(\lambda \right)$ and $Y=I_{[0,a]}\left(X\right)X+I_{(a,\infty )}\left(X\right)a$ .

1. Use the fact that
   $${\int }_0^{\infty }t{e}^{-\lambda t}dt=\frac{1}{{\lambda }^2}\text{and}{\int }_a^{\infty }t{e}^{-\lambda t}dt=\frac{1}{{\lambda }^2}{e}^{-\lambda a}(1+\lambda a)$$
   to determine an expression for $E\left[Y\right]$ .
2. Use the approximation method, with $\lambda =1/50,a=30$ . Approximate the exponential at 10,000 points for $0\le t\le 1000$ . Compare the approximate result with the theoretical result of part (a).

(See Exercise 1 from "Problems On Random Vectors and Joint Distributions", m-file npr08_01.m ). Two cards are selected at random, without replacement, from a standarddeck. Let X be the number of aces and Y be the number of spades. Under the usual assumptions, determine the joint distribution. Determine $E\left[X\right]$ , $E\left[Y\right]$ , $E\left[X^2\right]$ , $E\left[Y^2\right]$ , and $E\left[XY\right]$ .

(See Exercise 2 from "Problems On Random Vectors and Joint Distributions", m-file npr08_02.m ). Two positions for campus jobs are open. Two sophomores, three juniors,and three seniors apply. It is decided to select two at random (each possible pair equally likely). Let X be the number of sophomores and Y be the number of juniors who are selected. Determine the joint distribution for $\{X,Y\}$ and $E\left[X\right]$ , $E\left[Y\right]$ , $E\left[X^2\right]$ , $E\left[Y^2\right]$ , and $E\left[XY\right]$ .

(See Exercise 3 from "Problems On Random Vectors and Joint Distributions", m-file npr08_03.m ). A die is rolled. Let X be the number of spots that turn up.A coin is flipped X times. Let Y be the number of heads that turn up. Determine the joint distribution for the pair $\{X,Y\}$ . Assume $P(X=k)=1/6$ for $1\le k\le 6$ and for each k , $P(Y=j|X=k)$ has the binomial $(k,1/2)$ distribution. Arrange the joint matrix as on the plane, with values of Y increasing upward. Determine the expected value $E\left[Y\right]$ .