, 2
1 A discrete random variable, X, has a Poisson distribution with mean 4.
(a) Write down the value of Var (X ). [1]
(b) Find P(X = 3). [1]
(c) Find P(X 2 3). [2]
Another discrete random variable, Y, which is independent of X, has a Poisson distribution with
mean 5.
(d) Find P(X + Y G 9). [2]
2 In an experiment, the speed of an object, v ms -1 , at time t seconds, is recorded at intervals of
10 seconds from t = 0 to t = 80.
The summary statistics are as follows.
n = 8, Rt = 280, Rt2 = 14000, Rv = 876.28, Rv2 = 133310.2, Rtv = 43190
A linear function of t is used as a model for v.
(a) By first determining the equation of an appropriate regression line, estimate the value of v
when t = 45. [3]
(b) Comment on the reliability of your estimate in part (a). [1]
(c) Explain why in this scenario it would be inappropriate to use a value of v to estimate a value
of t using any regression line. [1]
© OCR 2025 Y432/01 Jun25
, 3
3 An archer fires arrows one by one at a target until an arrow hits the target. The number of arrows
fired up to and including the arrow that hits the target is denoted by X. The archer considers using
a geometric model for the distribution of X.
(a) State two assumptions required for such a model to be appropriate. [2]
For the remainder of this question you may assume that a geometric model is appropriate for the
distribution of X.
The archer keeps records of the values of X. He collects a sample of these values and uses them to
estimate the variance of X.
(b) Give two features that the sample should have. [2]
The estimate of the variance of X calculated from the sample is 12.
(c) Determine an estimate of the probability that when the archer fires a single arrow at the
target, the arrow hits the target. [3]
(d) Use your answer to part (c) to write down an estimate of the expected number of arrows that
the archer must fire at the target up to and including the first arrow that hits the target. [1]
4 The discrete random variable, X, has a uniform distribution over the values {5, 6, ..., n}.
(a) In this question you must show detailed reasoning.
In part (a) you should assume that n = 24.
(i) Using the formula E(X) = / xi pi , determine the value of E(X) by direct calculation. [3]
(ii) Use the substitution Y = X - 4 and the given formula in the formulae booklet to
determine Var (X ). [2]
1
(iii) Show that P(X 2 E(X )) = . [1]
2
1
(b) A student states that P(X 2 E(X )) = for all n H 24.
2
Explain whether the student is correct. [1]
© OCR 2025 Y432/01 Jun25 Turn over
, 4
5 Long-term data collected at a certain farm indicates that 30% of eggs that it produces are large and
the rest are categorised as medium.
Eggs are collected in batches of 4 and the number of large eggs in each batch, X, is recorded.
It is suggested that a binomial distribution might be appropriate to model X.
(a) (i) State two reasons, within the context of this question, why a binomial model might be
appropriate to model X. [2]
(ii) State two assumptions, within the context of this question, that must be made for a
binomial model to be appropriate to model X. You should not refer to the reasons given
in part (a)(i). [2]
(iii) Write down the binomial distribution which might be a suitable model for the
distribution of X. [1]
The farm manager investigates whether the distribution in part (a)(iii) is a good model for X.
(b) State one reason why taking data from a sample of 6 batches would not be appropriate to
carry out the investigation. [1]
The farm manager takes a random sample of 100 batches. The results are summarised in
Table 5.1.
Table 5.1
x 0 1 2 H 3
Frequency 31 36 30 3
Table 5.2, shows expected frequencies for the values of X in 100 batches, assuming that X has the
distribution from part (a)(iii). The table is incomplete.
Table 5.2
x 0 1 2 H 3
Expected frequency 24.01
(c) Complete the copy of Table 5.2 in the Printed Answer Booklet. Give the expected
frequencies to 2 decimal places. [2]
(d) Carry out a χ2 test, at the 5% significance level, for goodness of fit of the model in
part (a)(iii). [5]
© OCR 2025 Y432/01 Jun25
1 A discrete random variable, X, has a Poisson distribution with mean 4.
(a) Write down the value of Var (X ). [1]
(b) Find P(X = 3). [1]
(c) Find P(X 2 3). [2]
Another discrete random variable, Y, which is independent of X, has a Poisson distribution with
mean 5.
(d) Find P(X + Y G 9). [2]
2 In an experiment, the speed of an object, v ms -1 , at time t seconds, is recorded at intervals of
10 seconds from t = 0 to t = 80.
The summary statistics are as follows.
n = 8, Rt = 280, Rt2 = 14000, Rv = 876.28, Rv2 = 133310.2, Rtv = 43190
A linear function of t is used as a model for v.
(a) By first determining the equation of an appropriate regression line, estimate the value of v
when t = 45. [3]
(b) Comment on the reliability of your estimate in part (a). [1]
(c) Explain why in this scenario it would be inappropriate to use a value of v to estimate a value
of t using any regression line. [1]
© OCR 2025 Y432/01 Jun25
, 3
3 An archer fires arrows one by one at a target until an arrow hits the target. The number of arrows
fired up to and including the arrow that hits the target is denoted by X. The archer considers using
a geometric model for the distribution of X.
(a) State two assumptions required for such a model to be appropriate. [2]
For the remainder of this question you may assume that a geometric model is appropriate for the
distribution of X.
The archer keeps records of the values of X. He collects a sample of these values and uses them to
estimate the variance of X.
(b) Give two features that the sample should have. [2]
The estimate of the variance of X calculated from the sample is 12.
(c) Determine an estimate of the probability that when the archer fires a single arrow at the
target, the arrow hits the target. [3]
(d) Use your answer to part (c) to write down an estimate of the expected number of arrows that
the archer must fire at the target up to and including the first arrow that hits the target. [1]
4 The discrete random variable, X, has a uniform distribution over the values {5, 6, ..., n}.
(a) In this question you must show detailed reasoning.
In part (a) you should assume that n = 24.
(i) Using the formula E(X) = / xi pi , determine the value of E(X) by direct calculation. [3]
(ii) Use the substitution Y = X - 4 and the given formula in the formulae booklet to
determine Var (X ). [2]
1
(iii) Show that P(X 2 E(X )) = . [1]
2
1
(b) A student states that P(X 2 E(X )) = for all n H 24.
2
Explain whether the student is correct. [1]
© OCR 2025 Y432/01 Jun25 Turn over
, 4
5 Long-term data collected at a certain farm indicates that 30% of eggs that it produces are large and
the rest are categorised as medium.
Eggs are collected in batches of 4 and the number of large eggs in each batch, X, is recorded.
It is suggested that a binomial distribution might be appropriate to model X.
(a) (i) State two reasons, within the context of this question, why a binomial model might be
appropriate to model X. [2]
(ii) State two assumptions, within the context of this question, that must be made for a
binomial model to be appropriate to model X. You should not refer to the reasons given
in part (a)(i). [2]
(iii) Write down the binomial distribution which might be a suitable model for the
distribution of X. [1]
The farm manager investigates whether the distribution in part (a)(iii) is a good model for X.
(b) State one reason why taking data from a sample of 6 batches would not be appropriate to
carry out the investigation. [1]
The farm manager takes a random sample of 100 batches. The results are summarised in
Table 5.1.
Table 5.1
x 0 1 2 H 3
Frequency 31 36 30 3
Table 5.2, shows expected frequencies for the values of X in 100 batches, assuming that X has the
distribution from part (a)(iii). The table is incomplete.
Table 5.2
x 0 1 2 H 3
Expected frequency 24.01
(c) Complete the copy of Table 5.2 in the Printed Answer Booklet. Give the expected
frequencies to 2 decimal places. [2]
(d) Carry out a χ2 test, at the 5% significance level, for goodness of fit of the model in
part (a)(iii). [5]
© OCR 2025 Y432/01 Jun25
