7 Let U1 and U2 be two independent random variables, both uniformly

distributed over [0, a]. Let V = min{U1, U2} and Z = max{U1, U2}. Show

that the joint distribution function of V and Z is given by

F(s, t) = P(V ? s, Z ? t) = t

2 ? (t ? s)2

a2 for 0 ? s ? t ? a.

Hint: note that V ? s and Z ? t happens exactly when both U1 ? t and

U2 ? t, but not both s

9.18 Suppose a vase contains balls numbered 1, 2,...,N. We draw n balls

without replacement from the vase. Each ball is selected with equal probability,

i.e., in the first draw each ball has probability 1/N, in the second draw each

of the N ? 1 remaining balls has probability 1/(N ? 1), and so on. For i =

1, 2,...,n, let Xi denote the number on the ball in the ith draw. We have

shown that the marginal probability mass function of Xi is given by

pXi (k) = 1

N , for k = 1, 2,...,N.

a. Show that

E[Xi] = N + 1

2 .

b. Compute the variance of Xi. You may use the identity

1+4+9+ + N2 = 1

6

N(N + 1)(2N + 1).

9.19 Let X and Y be two continuous random variables, with joint probability density function

f(x, y) = 30

? e?50x2?50y2+80xy

for ??

a. Determine positive numbers a, b, and c such that

50x2 ? 80xy + 50y2 = (ay ? bx)

2 + cx2.

b. Setting = 4

5x, and ? = 1

10 , show that

(

?

50y ? ?

32x)

2 = 1

2

y ?

?

2

and use this to show that

?

??

e?(

?50y??32x)2

dy =

?2?

10 .

c. Use the results from b to determine the probability density function fX

of X. What kind of distribution does X have?

Find the angle between the lines whose slopes are 2

1 and 3

2. Find the distance of the point (4,1) from the line 3 4 x y - + 12 = 0

3. Show that the straight lines x y + - 4 0 = + ,3 2 x x = - 0 3 and 3 1 y + =6 0 are

concurrent.

4. Find the value of 'a' for which the straight lines 3 4 xy xy += -= 13;2 7 - 1 and

ax y - - 14 = 0 are concurrent.

5. A manufacturer produces 80 TV sets at a cost of `2,20,000 and 125 TV sets at a cost

of `2,87,500. Assuming the cost curve to be linear, find the linear expression of the

given information. Also estimate the cost of 95 TV sets.

3. The lifetime of a light bulb is modeled by an exponential random variable, ' A= 1 100 hours ' . What is the probability that the light bulb operates for more 3. The lifetime of a light bulb is modeled by an exponential random variable, with parameter A = Too hours . What is the probability that the light bulb operates for more than 500 hours?""""' ""J \"'J\"-"'J"'GJ""- "'"""'_'} Exercise 13. Suppose I tell you that the following list of 20 numbers is a random sample from a Gaussian random variable, but I don't tell you the mean or standard deviation. Also, around one or two of the numbers was corrupted by noise, computational error, tabulation error, etc., so that it is totally unrelated to the actual Gaussian random variable. 1.2045, 1.4329, O.3616, O.3743, 2.7298, 1.0601, 1.3298, 0.2554, 6.1365, 1.2185 2.7273, U.8453, 3.4282, 3.2270, 1.0137, 2.0553, 5.5393, 0.2572, 1.4512, 1.2347 To the best of your ability, determine what the mean and standard deviation are of this random variable. Supposing you had instead a billion numbers, and 5 or 10 percent of them were corrupted samples, can you come up with some automatic way of throwing out the corrupted samples? (Once again, there could be more than one right answer here; the question is intentionally open-ended.) 1. For each type of random variable described in Project 3.1 Step 1, calculate the theoretical mean and variance. Project 3.1 1. Generate a sequence of each of the following types of random variables; each sequence should be at least 10,000 points long- (a) A binomial random variable. Let the number of Bernoulli trials be n = 12. Recall that the binomial random variable is defined as the number of is in in trials for a Bernoulli (binary) random variable. Let the parameter p in the Bernoulli trials be p = 0.5109. (b) A Poisson random variable as a limiting case of the binomial random variable with p = 0.0125 or less and n = 80 or more while maintaining 0 = np = 1. (c) A type 1 geometric random variable with parameter p = 0.09. (d) A (continuous) uniform random variable in the range [-2, 5]. (e) A Gaussian random variable with mean / = 1.3172 and variance o' = 1.9236. (f) An exponential random variable with parameter A = 1.37