Page 200 - DMTH404_STATISTICS

Statistics

Notes The integral part of m will be mode.

Case II. When m is an integer
The distribution is bimodal with values m and m - 1.

Example 13: The average number of customer arrivals per minute at a super bazaar is 2.
Find the probability that during one particular minute (i) exactly 3 customers will arrive, (ii) at
the most two customers will arrive, (iii) at least one customer will arrive.
Solution.
It is given that m = 2. Let the number of arrivals per minute be denoted by the random variable
r. The required probability is given by

e  2 .2 3 0.13534 8
´
(i) P (r = ) 3 = = = 0.18045
3! 6
2 e  2 .2 r 4 ù
+
´
(ii) P (r  ) 2 = å = e  2 é ê 1 2 + ú = 0.13534 5 = 0.6767.
r= 0 ! r ë 2 û
e  2 .2 0


(iii) P (r  ) 1 = 1 P (r = ) 0 =  = 1 0.13534 = 0.86464.
1
0!
Example 14: An executive makes, on an average, 5 telephone calls per hour at a cost
which may be taken as Rs 2 per call. Determine the probability that in any hour the telephone
calls' cost (i) exceeds Rs 6, (ii) remains less than Rs 10.

Solution.
The number of telephone calls per hour is a random variable with mean = 5. The required
probability is given by

3 e  5 .5 r
1

(i) P (r > ) 3 = 1 P (r  ) 3 =  å
r= 0 ! r


1 e=   5 é 1 5 + 25 + 125 ù = 1 0.00678 ´ 236 = 0.7349.
+
ê ú
ë 2 6 û 6
4 e  5 .5 r 25 125 625ù 1569
(ii) P (r  ) 4 = å = e  5 é ê 1 5+ + + ú = 0.00678 ´ = 0.44324.
+
r= 0 ! r ë 2 6 24 û 24
Example 15: A company makes electric toys. The probability that an electric toy is
defective is 0.01. What is the probability that a shipment of 300 toys will contain exactly 5
defectives?

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