Download Tutorial Chapter 2

Probability and Statistics (EQT 272)
Tutorial Chapter 2
1. Determine whether the following random variables are discrete or continuous.
a)
b)
c)
d)
The number of eggs that a hen lays in a day.
The amount of milk a cow produces in one day.
The cost of making a randomly selected movie.
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.
 x2 5 
f ( x)  c    for x  0,1, 2,3, 4
 2 2
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”.
a) List the possible outcomes (the elements in the sample space S)
b) Let X be a random variable giving the number of “yellow” marbles. List the
outcomes for the random variable X.
c) Find the probability for each value of X.
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
P(X=x)
0
0.502
1
0.365
2
0.098
3
0.023
4
0.011
5
0.001
a) Verify whether this distribution is a probability distribution.
b) Find P(X ≥ 2)
c) Find P(0<X<4)
1
5. Let the probability density function of a random variable Y be
0  y 1
1 y  2
otherwise
,
y

f ( y )  2  y ,
0
,

a) Find F(Y)
b) Find P(0.5≤Y≤0.9)
c) Find P(0.75≤Y≤1.5)
6. Suppose X is a random variable with the following distribution function.
, x0
0

x3
 3
F ( x)  
(4 x 2  ) , 0  x  8
3
 256
, x8
1
a) Find P(-1≤ X ≤ 2)
b) Find f (x)
7. Let Y be a continuous random variable with the following probability density function.
, 0  y 1
y

f ( y )  2  y , 1  y  2
0
, otherwise

Calculate
a) E(Y)
b) Var (Y)
c)  Y

Answers:
1.
b)continuous
c) continuous
d) discrete
4
c)
2
55
5.

a) S  BB, BR, BY , RB , RR , RY , YB , YR, YY
a)
0,
 y2
 ,
2
F Y   
2 y 

1,

4.
9
9
b) 0.28
6.
b)
2
 1, for 1  y  2
2
7.
b) 0.133
c) 0.486
c) 0.594
a) 0.156
for 0  y  1
for y  2
1
,
a) probability distribution
for y  0
y
4
,
9
2.
3.

b) X  0, 1, 2
a) discrete
a) 1
 3
 8 x  x 2  , for 0  x  8
f  x    256

elsewhere
0,
b) 0.167
c) 0.408
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