Expectation and variance

The random variable G has a continuous uniform distribution over the interval [– 3, 15]

Calculate P(G 12)

The random variable H has a continuous uniform distribution over the interval [2, w]

The random variables G and H are independent and E(H) = 10

Show that the probability that G and H are both greater than 12 is

The random variable A is the area on a coordinate grid bounded by

y = –3

y = –4 | x | + k

where k is a value from the continuous uniform distribution over the interval [5, 10]

Calculate the expected value of A

The discrete random variable X has probability distribution

where a and b are probabilities.

Find E(X)

Given that Var(X) = 3.9

find the value of a and the value of b

The independent random variables X₁ and X₂ each have the same distribution as X

Find P X₁ X₂ 3

In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.

The results are summarised in the following table.

Marcus believes that a B(5, 0.5) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows

Find the value of r and the value of s

Carry out a suitable test, at the 5% level of significance, to determine whether or not the B(5, 0.5) distribution is a good model for these data. You should state clearly your hypotheses, the test statistic and the critical value used.

Nima believes that a better model for these data would be B(5, p)

Find a suitable estimate for p

To test her model, Nima uses this value of p, to calculate expected frequencies as follows

The test statistic for Nima’s test is 1.62 (to 3 significant figures)

(i) giving your reasons, the degrees of freedom (ii) the critical value that Nima should use for a test at the 5% significance level.

(i) the probability of the coin landing on heads, (ii) the independence of the spins of the coin. Give reasons for your answers.

There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die.

Estimate the probability that the mean number of rolls for the class is less than 27.2

The discrete random variable X has probability generating function

Specify the distribution of X

A fair die is rolled repeatedly.

Describe an outcome that could be modelled by the random variable X

Use calculus and G_X (t) to find

E(X)

Var(X)

The discrete random variable Y has probability generating function

Find the exact value of P(Y = 19)

The random variable R has a continuous uniform distribution over the interval [2, 10]

Write down the probability density function f(r) of R

A sphere of radius R cm is formed.

The surface area of the sphere, S cm², is given by S = 4πR²

The volume of the sphere, V cm, is given by V ϖ R³

Find, using algebraic integration, the expected value of V

The discrete random variable X has probability distribution

where b is a constant and b 5

Find E(X) in terms of b

Given that Var(X) = 34.26

find the value of b

Find P(X² < 2 – 3X)

A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

The discrete random variable V has probability distribution

Show that the probability generating function of V is

The discrete random variable W has probability generating function

Use calculus to find

E(W)

Var(W)

Given that V and W are independent,

find the probability generating function of X = V + W in its simplest form.

The discrete random variable Y = 2X + 3

Find the probability generating function of Y

Find P(Y = 15)

A rectangle is to have an area of 40 cm²

The length of the rectangle, Lcm, follows a continuous uniform distribution over the interval [4, 10]

Find the expected value of the perimeter of the rectangle. Use algebraic integration, rather than your calculator, to evaluate any definite integrals.

A courier delivers parcels. The random variable X represents the number of parcels delivered successfully each day by the courier where X ~ B (400, 0.64)

A random sample X₁, X₂,... X₁₀₀ is taken. Estimate the probability that the mean number of parcels delivered each day by the courier is greater than 257

Members of a photographic group may enter a maximum of 5 photographs into a members only competition. Past experience has shown that the number of photographs, N, entered by a member follows the probability distribution shown below.

show that Var(N) = 2.76

The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of £1 per person. Thus a member who enters 3 photographs pays 90p and a member who enters 4 or 5 photographs just pays £1

Assuming that the probability distribution for the number of photographs entered by a member is unchanged,

calculate the expected entry fee per member.

Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.

State a limitation of the Poisson distribution in this case.

Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red. Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.

Find the probability that Asha gets a red counter for the second time on her 9th draw.

The probability of Davinda getting a red counter on any one draw is p. Davinda draws counters until she gets n red counters. The random variable D is the number of counters Davinda draws.

Given that the mean and the standard deviation of D are 4400 and 660 respectively,

find the value of p.

Jerry believes that his bag contains a smaller proportion of red counters than Asha’s bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable J to be the number of counters drawn up to and including the first red counter.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test.

Jerry gets a red counter for the first time on his 34th draw.

Giving a reason for your answer, state whether or not there is evidence that Jerry’s bag contains a smaller proportion of red counters than Asha’s bag.

Given that the probability of Jerry getting a red counter on any one draw is 0.011

show that the power of the test is 0.702 to 3 significant figures.

The probability generating function of the random variable X is

where k is a constant.

Find P(X = 2)

Find the probability generating function of W = 2X + 3

The probability generating function of the random variable Y is

Given that X and Y are independent,

find the probability generating function of U = X + Y in its simplest form.

Use calculus to find the value of Var(U)

The continuous random variable X has cumulative distribution function given by

Find P(X < 5X 8)

Find the median of X.

Find E(X²)

Sketch the probability density function of X.

Describe the skewness of the distribution of X.

The discrete random variables W, X and Y are distributed as follows

W ~ B(10, 0.4) X ~ Po(4) Y ~ Po(3)

Explain whether or not Po(4) would be a good approximation to B(10, 0.4)

State the assumption required for X + Y to be distributed as Po(7)

Given the assumption in part (b) holds,

find P(X + Y < Var(W))

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Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.

Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

Find the probability that Suzanne selects the blue counter on her 4th selection.

Find the probability that the blue counter is first selected on or after Jon’s third selection.

Find the mean and standard deviation of the number of selections made until the blue counter is selected.

Find the probability that Suzanne wins the game.

The discrete random variable X has the following probability distribution.

Find Var(X)

The discrete random variable Y is defined in terms of the discrete random variable X

When X is negative, Y = X²

When X is positive, Y = 3X – 2

Find P(Y < 9)

Find E(XY)

A discrete random variable X has probability generating function given by

where a and b are positive constants.

Write down the value of P(X = 3)

find P(X = 2)

find E(X)

The random variable Y = 3X + 2

Find the probability generating function of Y

A six-sided die has sides labelled 1, 2, 3, 4, 5 and 6

The random variable S represents the score when the die is rolled.

Alicia rolls the die 45 times and the mean score, , is calculated.

Assuming the die is fair and using a suitable approximation,

find, to 3 significant figures, the value of k such that P( < k) = 0.05

Explain the relevance of the Central Limit Theorem in part (a).

Alicia considers the following hypotheses:

H₀: The die is fair H: The die is not fair

Given that the true distribution of S has mean 4 and variance 3

find the power of this test.

Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die. Give a reason for your answer.

The random variable X has probability density function f (x) and Figure 1 shows a sketch of f (x) where

The random variable Y~ N (μ, σ²) and E(Y) = E(X)

The probability density function of Y is g (y), where

Given that g (μ) = f (μ)

find the exact value of σ

A circle, centre O, has radius xcm, where x is an observation from the random variable X which has a rectangular distribution on [0, π ]

Find the probability that the area of the circle is greater than 10 cm²

State, giving a reason, whether the median area of the circle is greater or less than

The triangle OAB is drawn inside the circle with OA and OB as radii of length xcm and angle AOB x radians.

Use algebraic integration to find the expected value of the area of triangle OAB. Give your answer as an exact value.

The continuous random variable has cumulative distribution function given by

where k is a constant.

Showing your working clearly, use calculus to find

E(X)

the mode of X

Describe, giving a reason, the skewness of the distribution of X