Part A:
Given that Y is the number of phone calls received in a day and that a<span> new iPhone user receives an average of 4.2 phone calls per hour.
Thus, the average number of phone calls received in a day is 4.2 * 24 = 100.8
Therefore,
the </span><span>
λ for Y is 100.8Part B:
The standard deviation of a Poisson distribution is given by
Thus, the standard deviation of X is given by
Therefore,
the standard deviation of X is 2.05.
Part C:
The </span><span>standard deviation of Y is given by
Therefore,
the standard deviation of X is 10.04.
Part D:
</span>T<span>he probability of a Poisson distribution is given by
where, x is the number of occurence of the event in the given period.
Thus, the </span><span>probability that no calls are received in an hour is given by
Therefore, </span><span>
the <span>
probability that no calls are received in an hour is 0.015Part E:
We approximate the Poisson distribution using Normal distribution by noting that for a Poisson distribution
The probability of a normal distribution is given by
Thus, </span></span><span>the probability that more than 120 calls are received in a day is given by
Therefore,
the probability that more than 120 calls are received in a day is 0.0279
Part F:
</span><span>We also approximate this using Normal distribution
The probability of a normal distribution between two value (a, b) is given by
Thus, </span>the probability that Y is between 90 and 110 inclusive<span> is given by
Therefore,
the probability that Y is between 90 and 110 is 0.67922
</span>
Part G:
<span>The variance of aY is given by
Thus, the variance of
is given by
Therefore, </span><span>
the variance of is 201.6Part H:
</span><span>We approximate this using Normal distribution
The probability of a normal distribution is given by
Thus, </span>the probability that 120 or fewer calls<span> are received in a day is given by
Since, </span><span>this distribution holds for all hours and days and one call is independent from another. Assuming that there are seven days in a week.
</span>T<span>he probability that 120 or fewer calls are received for every day of this week is given by
</span><span>Therefore,
the probability </span><span>
<span>that 120 or fewer calls are received for every day of this week is</span> 0.8202</span>