Statistics and Probability Answers

A6. A hospital receives 20% of its COVID-19 vaccine shipments from Ghana and the
remainder of its shipments from neighbouring countries. Each shipment contains a very
large number of vaccine vials. For Ghana’s shipments, 10% of the vials are ineffective.
For the neighbouring countries, 2% of the vials are ineffective. The hospital tests 30
randomly selected vials from a shipment and finds that one is ineffective. What is the
probability that the shipment came from Ghana.

A5. A baseball team has scheduled its opening game for April 1. It is assume that if it
snows on April 1, the game is postponed and will be play on the next day that, it does
not snow. The team purchased insurance against snow. The policy will pay GHS 1,000
for each day, up to 2 days that the game is postponed. It is determined that the number
of consecutive days of snow beginning on April 1, is a Poisson random variable with
mean 0.6. What is the standard deviation of the amount that the insurance company
will have to pay.
A6. A hospital receives 20% of its COVID-

A4. The loss due to an earthquake in a commercial building is modelled by a random
variable X with density function
f(x) = 
0.005(20 0 x), for 0 < x < 20
0, elsewhere
Given that the fire loss exceeds 10, what is the probability that it exceeds 18

A3. The lifetime of a machine is continuous on the interval (0, 40) with probability density
function f, where f(t) is proportional to (t + 10))2
, and t is the lifetime in years.
Calculate the probability that the lifetime of the machine part is less than 10 years.
Hint: Show that f(t) is legitimate and find the proportionality constant.

A2. A random variable X has the cumulative distribution function given as
F(x) =



0, for x < 1
x
2 2 2x + 2
2
, for 1 ≤ x < 2
1, for x ≥ 2
Calculate the variance of X.

A1. Let X be a continuous r.v with density function
f(x) = ( |x|
10
, for r 2 < x < 4
0, elsewhere
. Calculate the expected value of X.

The lifetime of a machine is continuous on the interval (0, 40) with probability density function f, where f(t) is proportional to (t + 10)−2, and t is the lifetime in years. Calculate the probability that the lifetime of the machine part is less than 10 years. Hint: Show that f(t) is legitimate and find the proportionality constant.

A baseball team has scheduled its opening game for April 1. It is assume that if it snows on April 1, the game is postponed and will be play on the next day that, it does not snow. The team purchased insurance against snow. The policy will pay GHS 1,000 for each day, up to 2 days that the game is postponed. It is determined that the number of consecutive days of snow beginning on April 1, is a Poisson random variable with mean 0.6. What is the standard deviation of the amount that the insurance company will have to pay.

A tobacco company produces blends of tobacco, with each blend containing various proportions of Turkish, domestic, and other tobaccos. The proportions of Turkish and domestic in a blend are random variables with joint density function (X = Turkish and Y = domestic)
f(x,y)={(3x-y)/9, 1<x<3 , 1<y<2,
0, elsewhere

a) Find the marginal density functions of X and Y.
b) Find the mean and variance of X and Y
c) Are X and Y independent?

8. A financial analyst has found out that policyholders are four times as likely to file two
claims as to file four claims. If the number of claims filed has a posson distribution,
what is the variance of the number of claims filed.

The lifetime of a machine is continuous on the interval (0, 40) with probability density
function f, where f(t) is proportional to (t + 10)) 2
, and t is the lifetime in years.
Calculate the probability that the lifetime of the machine part is less than 10 years.
Hint: Show that f(t) is legitimate and find the proportionality constant