<PRE><B>Question 1132181</B>
Let random variable x represent the number of heads when a fair coin is
tossed three times.
 
 (a) Construct a table describing the probability distribution. 
&lt;pre&gt;

Before making the table we have to do some other things first.

List the sample space, which is the list of all ways the coins 
can land:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Each of those are equally likely and since there are 8 of them
each has probability 1/8, which is 0.125 as a decimal.

There is only 1 case with 0 heads, TTT, which has probability 1/8.
That&#39;s the case when X=0 and P(X)=P(0)=1/8=0.125

There are 3 cases with 1 heads, HTT, THT, TTH, So the probability
of 1 head is 3 times 1/8 or 3/8 or 0.375 That&#39;s the case when X=1 
and P(X)=P(1)=3/8=0.375.

There are 3 cases with 2 heads, HHT, HTH, HHT, So the probability
of 2 heads is also 3 times 1/8 or 3/8 or 0.375 That&#39;s the case 
when X=2 and P(X)=P(2)=3/8=0.375.

There is only 1 case with 3 heads, HHH, which has probability 1/8.
That&#39;s the case when X=3 and P(X)=P(3)=1/8=0.125.

Now we can make the table:

 X   P(X)
 --------
 0   0.125
 1   0.375
 2   0.375
 3   0.125


That&#39;s the answer to (a).
&lt;/pre&gt;
(b) Determine the mean &lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; and standard deviation &lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt; of x. 
&lt;pre&gt;
We make a new table, starting with the same first column.

  X
  -----------------
  0
  1
  2
  3
  ----------------

Under the bottom line we determine the mean by averaging the 4
numbers of heads in all the rolls. We add the column and divide
by 4:

  X
  -----------------
  0
  1
  2
  3
  ----------------  
4)6.0
  1.5

the mean = &lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = (0+1+2+3)÷4 = 6÷4 = 6/4 = 3/2 = 1.5
So the mean = &lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = 1.5. Next we subtract the mean of 1.5 from
each of the values of X:
For X=0, we subtract 1.5 from 0 and get X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;=0-1.5 = -1.5
For X=1, we subtract 1.5 from 1 and get X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;=1-1.5 = -0.5
For X=2, we subtract 1.5 from 2 and get X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;=2-1.5 = +0.5
For X=3, we subtract 1.5 from 3 and get X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;=3-1.5 = +1.5

Then we list those in a new column out beside the first:

  X      X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;
  -----------------
  0     -1.5
  1     -0.5
  2     +0.5
  3     +1.5
  ----------------  
4)&lt;u&gt;6.0&lt;/u&gt;         
  1.5

Next we square each of the values of X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;:
For X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = -1.5, we square and get (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²=(-1.5)²= 2.25
For X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = -0.5, we square and get (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²=(-0.5)²= 0.25
For X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = +0.5, we square and get (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²=(+0.5)²= 0.25
For X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt; = +1.5, we square and get (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²=(+1.5)²= 2.25
Then we list those in a new column out beside the second:

  X      X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;     (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)&lt;u&gt;6.0&lt;/u&gt;                 
  1.5

Under the bottom line we determine the variance by averaging the 4
values of (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)² in all the rolls. We add the column and divide
by 4.

&lt;font size=4&gt;&lt;b&gt;NOTICE: We divide by 4 because this is a POPULATION, not a SAMPLE.
If it were a SAMPLE, we would divide by 1 less than 4, or 3.  But
this is the ENTIRE POPULATION of ALL the ways the coin can land on
the three tosses, so we divide by 4, not 3:&lt;/font&gt;&lt;/b&gt;

  X      X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;     (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)&lt;u&gt;6.0&lt;/u&gt;          4)&lt;u&gt;5.00&lt;/u&gt;         
  1.5=&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;          1.25=&lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;²=the variance

Finally we find the standard deviation, &lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;, by taking the
square root of the variance &lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;². The square root of 1.25
is approximately 1.118033989, so that&#39;s the standard 
deviation &lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;.  Round that off 
as your teacher told you.

  X      X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;     (X-&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;)²
  ---------------------
  0     -1.5     2.25  
  1     -0.5     0.25
  2     +0.5     0.25
  3     +1.5     2.25
  -------------------  
4)&lt;u&gt;6.0&lt;/u&gt;          4)&lt;u&gt;5.00&lt;/u&gt;        
  1.5=&lt;font face=&quot;symbol&quot;&gt;m&lt;/font&gt;          1.25=&lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;²=the variance
                 1.118033989=&lt;font face=&quot;symbol&quot;&gt;s&lt;/font&gt;=the standard deviation

Edwin&lt;/pre&gt;
  </PRE>