Linear Combinations Exercises

Exercises

1. Consider the following two joint probability distributions. Distribution 1:

   		[math]Y:[/math]
   Probabilities		[math]-1[/math]	[math]0[/math]	[math]1[/math]
   [math]X:[/math]	[math]-1[/math]	[math]0.0[/math]	[math]0.1[/math]	[math]0.0[/math]
   	[math]0[/math]	[math]0.2[/math]	[math]0.3[/math]	[math]0.3[/math]
   	[math]1[/math]	[math]0.0[/math]	[math]0.0[/math]	[math]0.1[/math]

   Distribution 2:

   		[math]Y:[/math]
   Probabilities		[math]-1[/math]	[math]0[/math]	[math]1[/math]
   [math]X:[/math]	[math]-1[/math]	[math]0.02[/math]	[math]0.10[/math]	[math]0.08[/math]
   	[math]0[/math]	[math]0.06[/math]	[math]0.30[/math]	[math]0.24[/math]
   	[math]1[/math]	[math]0.02[/math]	[math]0.10[/math]	[math]0.08[/math]

   In each case,

   1. find the marginal probability distributions of [math]X[/math] and [math]Y[/math];

   2. find out whether [math]X[/math] and [math]Y[/math] are independent.

2. You are an investment consultant employed by an investor who intends to invest in the stock market or in a deposit account with a building society. The percentage annual rate of return for the stock market is denoted by the random variable [math]S[/math]. For simplicity we assume that this rate of return will be one of four values: -10%, 0%,10% or 20%. The annual rate of interest on the deposit account (denoted [math]R[/math]) will be 4%, 6% or 8%. From previous experience, you believe that the joint probability distribution for these variables is:

   		[math]S:[/math]
   Probabilities		[math]-10[/math]	[math]0[/math]	[math]10[/math]	[math]20[/math]
   [math]R:[/math]	[math]4[/math]	[math]0[/math]	[math]0[/math]	[math]0.1[/math]	[math]0.1[/math]
   	[math]6[/math]	[math]0[/math]	[math]0.1[/math]	[math]0.3[/math]	[math]0.1[/math]
   	[math]8[/math]	[math]0.1[/math]	[math]0.1[/math]	[math]0.1[/math]	[math]0[/math]

   1. 1. Find the marginal probability distributions for [math]R[/math] and [math]S[/math]. What is the overall probability that the rate of return for the stock market will be positive? What is the probability that the rate of return for the stock market will exceed that from the building society deposit account?

      2. Calculate the mean and variance using each of these marginal distributions. What does this information imply about the relative merits of the two types of investment?

   2. Calculate the (population) covariance and (population) correlation between [math]R[/math] and [math]S[/math]. How would you interpret the value of the correlation?

   3. 1. One proposal you make to the investor is to split her savings equally between the stock market and the building society account. Find (using any appropriate method) the mean and variance of the random variable

         [math]A=0.5R+0.5S.[/math]

      2. Why might the investor prefer the proposed new 50/50 strategy to the simple strategies of investing all her money in the building society or in the stock market?

3. The random variables [math]X[/math] and [math]Y[/math] have [math]\mu _{X}=10[/math], [math]\sigma_{X}=3,\mu _{Y}=-1,\sigma _{Y}=4[/math].

   1. Find the mean and standard deviation of [math]V=X+Y[/math] when

      1. [math]X[/math] and [math]Y[/math] are independent;

      2. [math]\sigma _{XY}=-8[/math].

   2. Find the mean and standard deviation of

      [math]W=3X-2Y+8[/math]

      when [math]X[/math] and [math]Y[/math] are independent.

4. In the manufacture of disposable syringes, the manufacturing process produces cylinders of diameter [math]X_{1}\sim N\left( 20.2,0.04\right) [/math] and plungers of diameter [math]X_{2}\sim N\left( 19.7,0.0225\right)[/math].

   1. If the components are combined so that [math]X_{1}[/math] is independent of [math]X_{2}[/math], what proportion of plungers will not fit?

   2. Suppose now that the components are not independently combined, but that larger plungers tend to be combined with larger cylinders, leading to [math]Cov\left[ X_{1},X_{2}\right] =0.02[/math]. What proportion of plungers will not fit now?