Question

4. Assume that the length of time between charges of a particular cell phone is normally distributed with a mean of 8 hours and a standard deviation of 2 hours. Find the probability that the cell phone will last between 5 and 10 hours between charges.

Answer 1

4.) We know that standard normal curve is symmetric about zero it means area on both sides of mean(0) is equally divided that is 0.5.

It means we can write

P(Z<z)=0.5 + P(0<Z<z)

P(Z>z)=0.5 - P(0<Z<z)

P(Z<-z)=P(Z>z)

The required probability is 0.7745

5.) Mean is 8/3.

6.) Poisson distribution occurs when there are events which do not occur as outcomes of a definite number of trials(unlike that in binomial distribution) of an experiment but which occur at random in some point of time and space wherein our interest lies only in the number of occurrences of the event, not in it's non-ocurrances.

In our question we are observing number of visitors (occurances) in 1 hour from 12:00pm to 1:00pm therefore our random variable X, number of visitors is following poission distribution.

We know that p.m.f of Poisson distribution with mean $\lambda$ is defined as

$P(X=x)=\frac{e^{-\lambda}\lambda^{x}}{x!}$

Our $\lambda$ is 29 then

e -29 29" P(X = x) c!

Now probability of exactly 20 visitors is given by

$P(X=20)=\frac{e^{-29}29^{20}}{20!}$

Sop 4.) Given Mean = 8 standard deviation =2 net x : length of the time between cherzzes of cellphone a criven XN Now N(o, 2² N(8,4) the probability that the cell phone will last between 5 and 10 hours between charges is given by PCsLX clo) = P(5-8 < X < 10-8) = P(-1.52Z < 1) P(221) - P(Z 2-1.5) 3 0.57 Ploczet) - P(2715) = 0:8 +POOLZZI) 20.5- Plocz< 1:5)? PCOLZL15) +Pfotz410) 0.4332 + 0.3413 0.7745 5. 0 1

5.) Given Continuous dandom Variable x 8 > ocx 24 f(x) 0 otherwe Now , We know that Mean, 4 = E(x) AO Jx. of (x) dx 4 fox) is zero s. Jx. 3 dx & 1 2 3 4 n $ 64 3 o] tera 8 3 G. average Given the information 1:00 pm number of visitors to the between and 12:00 pm booth is 29. we are observing In this situation occurance of event in some unit f time henie random vanlable x, number of visitors is following poission distribution with average 29. P. m.f of Possron (d) PCX 32) edd Xi Now. P(x=) é 6297 (S -29 20 않아