Download Probability and Statistics EQT 272

Probability and Statistics EQT 272
Semester 2
2013/2014
TUTORIAL 2
1) Determine whether the following random variables are discrete or continuous.
i. The number of eggs that a hen lays in a day.
ii. The amount of milk a cow produces in one day.
iii. The cost of making a randomly selected movie.
iv. The number of goals scored by a randomly selected football player in a soccer
tournament.
2) Determine the value c so that the following function is a probability function for a
discrete random variable.
f ( x)  c(
x2 5
 ) for x  0,1,2,3,4
2 2
ans: 2/55
3) A box contains three marbles (one blue, one red and one yellow). Two marbles are
drawn with replacement. This means a marbles is selected, its colour is observed and
then it is replaced in the box. A second marble is then selected and its colour is
observed. Let B denotes “blue” , R denotes “red” and Y denotes “yellow”.
i. List the possible outcomes (the elements in the sample space S)
ans: S = {BB, BR, BY, RB, RR, RY, YB, YR, YY}
ii. Let X be a random variable giving the number of “yellow” marbles. List the outcomes
for the random variable X.
ans: X={0, 1, 2}
iii. Find the probability for each value of X.
ans: 4/9, 4/9, 1/9
4) A factory manufactures DVDs. Batches of DVDs are randomly selected. The number
of defects (X) for each batch is observed and the following distribution is obtained.
X
0
1
2
3
4
5
P(X=x)
0.502
0.365
0.098
0.023
0.011
0.001
i. Verify whether this distribution is a probability distribution.
ii. Find P(X ≥ 2)
ans: 0.133
iii. Find P(0<X<4)
ans: 0.486
Probability and Statistics EQT 272
Semester 2
2013/2014
iv. Find the E(X), Variance(X) and standard deviation X.
5) Let the following function of a random variable Y be
0  y 1
1 y  2
otherwise
 y,

f ( y )  2  y ,
0

i. Check whether the function is probability density function
ii. Find the cumulative distribution function of Y, F (Y)
iii. Find P(0.5≤Y≤0.9)
ans: 0.28
iv. Find P(0.75≤Y≤1.5)
ans: 0.594
6) If a random variable X has probability density function


B x 2  4 x  5
f ( x)  
0

calculate the value of B.
0 x5
x  0 or x  5
ans: 3/50
7) Suppose X is a random variable with the following distribution function.
x0
0,

3
x
 3
F(X )  
(4 x 2  ),
0 x8
3
 256
x 8
1,
Find P(-1≤ X ≤ 2)
i.
ans: 0.156
ii.
Find f (x)
8) Let Y be a continuous random variable with the following probability density function.
0  y 1
1 y  2
otherwise
 y,

f ( y )  2  y,
0,

Calculate
i.
expected value of Y, E(Y)
ans: 1
ii.
variance of Y, Var (Y)
ans: 1/6
iii.  Y
ans: 0.408