Quiz 1

Justify all answers with neat and organized work. Clearly indicate your answers. 20 points possible.

1. (6 pts.) Here are some formulas.

x   = 1 n n å i=1  xi       u   = 1 n k å i=1  fi ui      s2 = n å i=1  xi2-n x   2   n-1       su2= k å i=1  fi ui2-n u   2   n-1

Use appropriate formulas to find the sample mean and sample variance for the given grouped sample data. Show your work.

Class Inteval        Class Mark (ui)        Frequency (fi)    (41.5, 45.5) 43.5 25 (45.5, 49.5) 47.5 14 (49.5, 53.5) 51.5 7

2. (6 pts.) Consider a random variable W which has p.d.f.

f(w)= 3 8 w2,       0 £ w < 2.

Use the formulas

m = ó õ S  xf(x) dx

and

s2= ó õ S  x2 f(x) dx -m2

to find the mean and variance of this distribution. Also find p_0.4, the 40th percentile.

3. (8 pts.) (a) Consider the geometric series

1 1-z =1+z+z2+z3+z4+...,        (-1 < z < 1).

Differentiate both sides of the above equation. Then multiply both sides of the resulting equation by z.

(b) A fair six-sided die is tossed successively until a 3 or higher is observed. Let the random variable X denote the number of tosses that are required. Then the space of X is S={x:x=1,2,3,4,...}. Later we learn that under certain conditions, we can assign probabilities to these outcomes in S with the function

f(x)=2 æ ç è 1 3 ö ÷ ø x   ,    x=1,2,3,4,....

Compute the mean m.

Hint: Use part (a) somewhere.