Practise calculating Poisson probabilities, rescaling the mean across time or space and choosing Poisson or normal approximations with conditions and continuity corrections.
Sales of cell phones at a certain shop occur singly, randomly and independently.
State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
The average number of sales per hour is 1.2 .
Assume now that a Poisson distribution is a suitable model.
Constant average rate
OE
Marking guidance:
Accept constant rate. Allow without context.
Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
=14.4
e^-14.4(14.4^1313!+14.4^1414!+14.4^1515!)
or e^-14.4(183837+189089+181526)
or (0.102469+0.105396+0.101181)
Poisson P(13,14,15). Expression must be seen.
Allow one end error;
allow any . Allow fully correct sigma
notation.
=0.309(3 sf)
SC: 0.309 with no working scores B1 B1.
Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1 -month period ( 140 hours) will be less than 150 .
N(140 x 1.2,140 x 1.2) or N(168,168)
Stated or implied.
149.5-168168[=-1.427]
Standardising using their mean and variance. Allow with wrong or no continuity correction.
( " -1.427 ") =1-( " 1.427 ")
For area consistent with their working.
=0.0768 or 0.0767 ( 3 sf )
The random variable W has a Poisson distribution.
State the relationship between E(W) and Var(W).
E(W)=Var(W).
Allow 'they are the same' OE.
Must be = not (and not both = and ).
Condone E(W)= and Var(W)=.
The random variable X has the distribution B(n, p). Jyothi wishes to use a Poisson distribution as an approximate distribution for X.
Use the formulae for E(X) and Var(X) to explain why it is necessary for p to be close to 0 for this to be a reasonable approximation.
n p n p(1-p), hence 1-p must be close to 1
OE. Must see formulae and q=1-p must be seen or
implied and conclusion made.
Given that Y has the distribution B(20000,0.00007), use a Poisson distribution to calculate an estimate of P(Y>2).
=1.4
Seen.
1-e^-1.4(1+1.4+1.4^22) or 1-e^-1.4(1+1.4+0.98) or 1- (0.2466+0.3452+0.2417)
1-e^-1.4(1+1.4+1.4^22) or 1-e^-1.4(1+1.4+0.98) or 1-
(0.2466+0.3452+0.2417)
Allow any ;
allow one end error. Expression must be
seen (accept correct sigma notation).
=0.167(3 sf) or 0.166
Use of Binomial scores SCB1 for 0.167 or 0.166.
No working: 0.167 [or 0.166] SC B1.
Note: =1.4 and 0.167 with no working seen scores SC
Use of Normal scores B0M0.