MATH 495
Problem Set 4
1. Let X be a random variable whose probability density function is f

1
2 e
  t

1

2
0 if 1
 t
 
. What is otherwise the expected value (mean) of X ?
2. Given i n 

1 i
2  n
 n

1

2 n

1

, what is the value of
6
75  i

50 i
2
?
3. The curves y

 sin x
 cos x

4
and y

 sin x
 cos x

4
intersect at x

0 . If a is the smallest positive value of x at which these curves next intersect, determine the exact area between these curves from x

0 to x
 a .
4. Let x
 r cos

and y
 r sin

. Find an expression for
 
 r
, the partial derivative of

with respect to r , keeping x constant.
5. Solve for x : x

1
 x

2 x

1

2 x

1
6.
7.
If a and x are positive numbers and A
 a
2
, express a log a x
 log
A x
in simplest form in terms of x .
The heights of the female employees in a large company are normally distributed with mean of 168 cm and standard deviation of 7 cm. What is the probability that a female employee selected at random will be taller than 182 cm?
8. Two points, x and y , are selected at random in the interval
  xy will be less than 1
2
?
9. In how many ways can the number 14 be written as the sum of 5 nonnegative integers? In how many ways can 14 be written as the sum of 5 positive integers?
10. You deal 5 cards to yourself from a standard deck. The result is 3 diamonds, a club and a spade. You discard the club and the spade, and draw two more cards from the deck. What's the probability of getting a flush (that is, of drawing two more diamonds)?
11. Determine the exact value of: arctan 1
 arctan 2
 arctan 3 .
12. Find n lim
  k n 

1

 k n
2
 k
2 n
3

 .
13. Find the area where the two circles pictured to the right intersect. y x

14. In the Taylor Series expansion (in powers of x ) of the function f
 
 e x
2  x
, what is the coefficient of x
3
?
15. Of all the integer pairs
100
 x

200
 
that satisfy the equation
. What is this integer pair?
42 x

55 y

1 , only one such pair has