Chapter 5: Probability and Random Variables

Expected Utility. One method for deciding among various investment involves the concept of expected utility. Economists describe the importance of various levels of wealth by using utility functions. For instance, in most case, a single dollar is more important (has greater utility ) for someone with little wealth than for someone with greater wealth Consider two investments, say investment A and B. Measured in thousand of dollars, suppose that investment A yields 0, 1, and 4 with probability 0.1 and16 with probability 0.5, 0.3 and 0.2 respectively. Let Y denote the yield of an investment. For the two investment, determine and compare.

Part (a) The mean of Y, the expected yield.

Part (b) The mean of $\sqrt{Y}$,the expected utility, using the utility function role="math" localid="1651845051902" $\upsilon \left(y\right)=\sqrt{y}$Interpret the utility function $u$

Part (c) The mean of $y^{3/2}$,the expected utility, using the utility function $v\left(y\right)=y^{3/2}$. Interpret the utility function v

Equipment breakdowns: A factory manager collected data on the number of equipment breakdown per day. From those data she derived the probability distribution shown in the following table, Where W denote the number of breakdown on a given day.

Part (a) Determine ${\mu }_{w}and{\sigma }_w$, Round your answer for the standard deviation to three decimal places.

Part (b) On average, how many breakdown occur per day?

Part (c) About how many breakdown are expected during a 1 year period assuming 250 work days per year?

Simulation. Let X be the value of a randomly selected decimal digit. that is a whole number between 0 and 9 inclusive.

Part (a) Use simulation to estimate the mean of X Explain your reasoning.

Part (b) Obtain the exact mean of X by applying definition 5.9 on page 22.7. Compare your result with that in part (a)

Queuing Simulation. Benny's Barber Shop in Cleveland has five chairs for waiting customers. The number of customers waiting is a random variable $Y$with the following probability distribution.

mass placed at point $x_k$on the seesaw. The center of gravity of these masses is defined to be the point $c$on the horizontal axis at which a fulcrum could be placed to balance the seesaw.

Relative to the center of gravity, the torque acting on the seesaw by the mass $p_k$is proportional to the product of that mass with the signed distance of the point $x_k$from c, that is, to $\left(x_k-c\right)·p_k$. Show that the center of gravity equals the mean of the random variable$X$. (Hint: To balance, the total torque acting on the seesaw must be 0 .)

Mean as center of Gravity. Let X be a discrete random variable with a finite number of number of possible values say $x_1,x_2.....x_m$For convenience, set $P_K=P(X=x_k),$for K = 1, 2.....m . Think of a horizontal axis as a seesaw and each $P_K$as a mass placed at point $x_k$on the seesaw. The center of gravity of those masses is defined to be the point c on the horizontal axis at which a fulcrum could be placed to balance the seesaw.

Relative to the center of gravity , The torque acting on the seesaw by the mass $P_K$is proportional to the product of that mass with the signed distance of the point $X_k$From c That is , to ($(x_k-c)$. $P_k$show that the center of gravity equal the mean of the random variable X ( hint: To balance, the total torque acting on the seesaw must be 0)

In probability and statistics, what is each repetition of an experiment called?

Poker Chips. A bowl contains 12 poker chips-3 red, 4 white, and 5 blue. If one of these poker chips is selected at random from the bowl, what is the probability that its color is

(a) red? (b). red or white? (c). not white?

Under what three conditions are repeated trials of an experiment called Bernoulli trials?

Explain the significance of binomial coefficients with respect to Bernoulli trials.

Discuss the pros and cons of binomial probability tables.