Assignment

1. Suppose that length of life in Japan, X, has exponential distribution: X~EXP(β). The pdf of X is given by:

f(x)={(βe^(-βx)&x≥0,@0&x<0.)¦

What is the support of X?

Prove that indeed, the above function is a pdf (i.e. nonnegative, and integrates to 1 over the entire support).

Show that the probability that a newborn will live at least 100 years is e^(-100β).

Suppose that only 5% of the newborns live more than the age ofx^*. Show that x^*=ln?0.05/(-β).

2. Consider the random experiment of tossing two dice.

Write the sample space for this random experiment.

Let X be a random variable which records the maximum of the two dice. List all the possible values of X (i.e., describe the support of X).

Show the probability density function of X.

Calculate the expected value (mean) of X.

Calculate the variance of X.

3. Let X be a continuous random variable, with pdf

f(x)={(1-0.5x&0≤x≤0,@0&"otherwise" .)¦

Verify that fis indeed a probability density function (i.e. it is nonnegative, and integrates to 1 over the entire support).

Using Excel, plot the graph of this pdf.

Calculate the mean of X.

Calculate the variance of X.

4. Let X be a random variable with mean μ and variance σ^2, and let Y=(X-μ)/σ be the "standardized" transformation of X.
Using rules of expected values show that the mean of Y is 0.

Using the rules of variances, show that the variance of Y is 1.

5. Consider the function

f(x)={(2-x-y&0≤x≤1, 0≤y≤1@0&"otherwise" .)¦

Show that is a probability density function.

Check whether and are statistically independent.

6. Let X be a random variables, and a,b be some numbers. LetY=ax+b be some linear transformation of X.Prove that: ifa>0, then corr(X,Y)=1, if if a<0, then corr(X,Y)=-1, and ifa=0, then corr(X,Y)=0.

7. Meteorologists study the correlation between humidity H, and temperature. Some measure the temperature in Fahrenheit F, while others use Celsius C, whereC=5/9(F-32).

Show that two researchers, who use the same data, but measure temperature in different units, will nevertheless find the same correlation between humidity and temperature. In other words, show that

Will the researchers get the same covariance if they use different units? Prove your answer.

Based on your answers to a and b, should researchers report covariance or correlation from their studies? Why?

8. Let X_1andX_2 be identically distributed random variables, and thus both have the same meanμ and variance σ^2. LetX ¯ be the average of X_1 and X_2, that is
X ¯=1/2 X_1+1/2 X_2

Show that the expected value (mean) of X ¯ isalso μ.

Find the variance ofX ¯.

Show that if X_1 and X_2 are independent, then the variance of X ¯isσ^2/2.

9. This question generalizes the previous one to average of any number of identically distributed random variables. LetX_1,...,X_2 be n identically distributed random variables with mean μ and variance σ^2. Let the average of these variables be
X ¯_n=1/n ∑_(i=1)^n X_i

Show that the mean of X ¯ is also μ.

Show that ifX_1,...,X_2are independent, then the variance ofX ¯ is σ^2/n.

What is the limit of var(X ¯ ) asn→∞, still assuming that X_1,...,X_2 are independent?