# Probability and Statistics 1

Probability and Statistics 1

Probability and Statistics 1

Section A — Answer all questions in this section.

Questions A1 to A20 require you to write down a statement or to perform a calculation.

Show the main steps, with brief explanation, in your answers.

When giving your answers in decimal form, you should use 4 decimal places.

If you use RStudio then you should state and describe the commands and options used.

A1. Events A and B are such that Pr(A) = 0.4, Pr(B) = 0.25 and Pr(A ∪ B) = 0.6. What

is Pr(Ac ∩ B)?

A2. Ben randomly chooses 4 people from a group of 7 friends to form a quiz team. How many

ways are there of forming the team?

A3. The owner of a fairground game estimates that about 1 in 100 of her customers will win

a prize worth £20, the others will win nothing. If she hopes to make a profit of 50p per

customer that plays, how much should she charge each customer to play?

A4. Maya is on an outdoor activity holiday. The probability that there are big enough waves

to be able to water surf on any given day is 0.3 and the size of waves on different days are

independent. What is the probability that it will be at least 3 days before Maya is able to

water surf?

A5. Let X be a random variable with mean −3 and variance 7 and Y be a random variable

with mean 6 and variance 3. Further, suppose Cov(X, Y ) = 0.3. What is the variance of

Z = 2X − Y ?

A6. In a game show, a single contestant randomly chooses to open one of 6 doors, one of

which leads to a prize. Out of 20 shows, what is the probability that exactly 3 contestants

will open the door leading to the prize?

A7. Suppose that X has a Poisson distribution with mean 2. What is the probability that X

is at least 2?

A8. Suppose that a continuous random variable X has a uniform distribution on the interval

from −1 to +1. What is the probability that X is between 2/3 and 4/3?

A9. Suppose the weight of a bag of rice can be modelled using a normal distribution with mean

1005g and standard deviation 20g. What is the probability that a randomly chosen bag

weighs less than 1000g?

A10. If a continuous random variable Z has a standard normal distribution, then what is

P r(Z ≥ 1.5 | Z > 0)?

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Module Code: MATH171001

A11. In RStudio, a vector mydata contains the values from a dataset of size 10. Write down

R commands to show (i) only the second element of the vector and (ii) all values except

the fourth and the sixth elements.

A12. If event A is favourable for event B, then prove that B is also favourable for A.

A13. If events A and B are such that A ⊂ B ⊂ Ω, then put the following in order of increasing

value (whenever possible):

Pr(A), Pr(A|B), Pr(B|A).

A14. In RStudio, a vector x contains the values in the range space of discrete random variable

X and a vector p contains the corresponding probabilities. Write down an R command to

evaluate E[X(X − 1)] from x and p.

A15. Considering random variable X, show that the following identity is valid:

E[(X − E[X])3

] = E[X3

] − 3E[X2

]E[X] + 2{E[X]}

3

.

Clearly state any standard results of expectation used.

A16. Suppose that the tossing of a fair coin leads to the following sequence of outcomes:

Head, Head, Tail. What question would you ask the experimenter to help you decide if

a binomial distribution or a geometric distribution was the most suitable model for this

situation?

A17. Let X be a beta random variable with parameters α = 1 and β = 2. Write down an R

command to evaluate the probability that X is more than 1/2.

A18. Suppose that X has the probability density function fX(x) = x

2/k for −1 ≤ x ≤ 1 and

fX(x) = 0 otherwise, where k is a constant. Write down a set of R commands to calculate

numerically the value of k.

A19. Assume that X1, X2, . . . , Xn are independent and identically distributed random variables

with expectation µ and finite variance σ

2

, and where n is large. State the approximate

distribution of Sn = X1 + X2 + · · · + Xn, including all parameters of the distribution.

A20. In a Bayesian analysis, suppose that a probability parameter p is to be modelled using a

beta distribution with parameters α and β, that is p ∼ Beta(α, β). What values of α and

β should be chosen to achieve a prior mean of 1/2 and a prior variance of 1/16?

Probability and Statistics 1

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