Question

Five coins were simultaneously tossed 1000 times and at each toss, the number of heads was observed. The number of tosses during which 0, 1, 2, 4, and 5 heads were obtained are shown in the table below. Convert the given frequency distribution to probability distribution find the expected value.

           Number of heads per toss:             0              1          2          3          4          5

           Number of tosses               :          38         144      342      287     164           25

Answer 1

Solution:

Step 1) Find mean of given frequency distribution and find estimate of p = probability of success

x : Number of heads per toss	f : Number of tosses	x*f
0	38	0
1	144	144
2	342	684
3	287	861
4	164	656
5	25	125

Thus mean is:

For Binomial distribution Mean is:

then

then

Step 2) Find initial probability:

Now use following recurrence relation to find probabilities

Thus we need to make following table:

In the following table, first we find (n-x)/(x+1) column by substituting n=5 and values of x from 0 to 5.

In the third column, we multiply second column by p^ /q^ =

for fourth column, P(X=x) , we put first probability as initial probability as it is obtained in step 2)

Then to get second probability we multiply first probability to first value of third column

that is: 0.033171 X 4.881425 = 0.161922 , which is second probability ( in second row)

then we multiply this second probability to corresponding second value of third column .

That is: 0.161922 X 1.952570 = 0.316164

Thus we get following table:

x	(n-x)/(x+1)	(n-x)/(x+1) *(p^/q^)	P(X=x)
0	5	4.881425	0.033171	33.171
1	2	1.952570	0.161922	161.9217
2	1	0.976285	0.316164	316.1635
3	0.5	0.488143	0.308666	308.6657
4	0.2	0.195257	0.150673	150.6729
5	0	0.000000	0.029420	29.41993

For last column, for expected frequencies : we multiply Each P(X=x) by total of frequency column = 1000

That is: 1000 X 0.033171 = 33.171

1000 X 0.161922 = 161.9217 and so on.