Dr. J's Maths.com
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Probability - Discrete distributions - Expected value and variance.
Test Yourself 1.


 

9. A random variable X has a probability distribution with E(X) = 22 and E(X2) = 565.

What is the standard deviation of X?

Calculate mean (expected value). 1. A probability distribution is summarised as:
x 0 1 4 9 16
Pr (X=x) 0.1 0.2 0.4 0.2 0.1

Determine the value of E(x).

Answer.5.2.		 2. A random variable X represents the number of cars crossing a particular intersection in a 10 minute interval.

The random variable is known to have the following probability distribution:

x 0 1 2 3 4
Pr (X=x) 0.05 0.2 0.3 0.4 0.05

What is the expected number of cars expected to cross the intersection in 10 minutes?

Answer.2.2 cars.
  3. In a fund-raising raffle for the school, 500 tickets are printed. First prize is $200 and there are four second prizes of $25 each.

(i) Summarise this information in a probability distribution table.

(ii) find the expected value of the winnings for each ticket.

Answer.$1.65.
  4. A book store keeps records about the number of book purchased by customers.

The store owner deduces from the data that the probability of customers purchasing no books is 0.5, of purchasing one book is 0.3 and of purchasing 2 books is 0.2.

What is the expected number of books purchased by customers?

Answer.0.7 books.
  5. You win a competition and you have a choice from two prizes:
  • $450 in cash; or
  • a gold coin.

The gold in the coin has a 40% chance of being worth $500, a 50% chance of being worth $400 and a 10% chance of being worth $200.

(i) Justify which option has the best financial benefit by determining the expected value from the probability distribution.

(ii) how could you vary the probability distribution (using realistic probabilities) to make the expected values of the prizes equal?

Answer.E(X) = $310.
  6. Ellen is considering buying shares in a gold mining company. She estimates she has:
  • a 20% probability of making a 20% profit on these shares;
  • a 50% probability of making a 10% profit; and
  • a 30% probability of making a 15% loss.

(i) What is Ellen's expected percentage rate of return by buying shares in this gold miner?

(ii) If Ellen purchased $2,400 of these shares, what would she estimate her profit or loss to be?

Answer.(i) Expected % is 0.06 = 6%.
(ii) Expected profit = $144.
  7. The following table, taken from an earlier Australian Census, shows the probability of the number of children under 15 years of age (X) in indigenous families living in a particular area in Western Australia.
X 1 2 3 4 5 6 7
Pr (x) 0.351 0.294 0.194 0.102 0.036 0.014 0.009

(i) Express the distribution in graphical form.

(ii) What is the most likely number of children under 15 years of age in an indigenous family in this region?

(iii) What is the probability of an indigenous family in this region having fewer than 4 children under 15 years of age?

(iv) What is the probability of an indigenous family in this region having more than 5 children under 15 years of age?

 
Calculate the standard deviation. 8. A random variable X has a probability distribution with a mean of 6 and
E(X2) = 52.

What is the standard deviation of X?

Answer.SD = 9.
9. A random variable X has a probability distribution with E(X) = 22
and E(X2) = 565.

What is the standard deviation of X?

Answer.SD = 9.
  10. A random variable X has a probability distribution with E(X2) = 25 and a standard deviation of 3.

What is the expected value of X?

Answer.SD = 4.
Calculate both the mean and the standard deviation. 11. The random variable x has the following probability distribution:
X 0 1 2 3 4
p(x) 0.1 0.2 0.4 0.2 0.1

What is the variance of the probability distribution?

Answer.Var = 1.2.
  12. The discrete random variable X has a probability distribution P(X):
X 1 2 3 4
P(X) 0.3 0.45 a b

For this distribution, E(X) = 2.

(i) Show that a = 0.2 and b = 0.05.

(ii) Find the standard deviation of the distribution.

Answer.SD = 0.84.
  13. Let X be a random variable with the following probability distribution:
X 1 2 3 4
p(x) 0.4 0.3 0.2 0.1

(i) Find E(X) and Var (X).

(ii) Is E(X) a possible value of X?

Answer.(a) E(x) = 2.0 and V(x) = 1.
(b) Yes E(x) is a possible value of X.
  14. Let X be a random variable with the following probability distribution:
X 5 10 15 20 25
p(x) 0.05 0.30 0.25 0.25 0.15

Find μ and σ.

Answer.(i) μ = 15.75.
(ii) σ = 5.76.
  15. Let X represent the number of times a Year 12 student visits the nearby Subway shop to buy lunch in a month. The following table shows the probability distribution of X:
X (no of visits) 0 1 2 3
P(x) 0.10 0.30 0.40 0.20

(i) Calculate the mean (μ) and standard deviation ( -) of this distribution.

(ii) What is the probability that the student visits Subway at least twice in a month?

(iii) Determine Pr(X ≤ 1.5).

(iv) Draw a graph of the probability distribution. Mark in the mean and the range bordered by μ ± σ.

Answer.(i) μ = 15.7.
(ii) σ = 0.81.
  16. In a game of chance, a player rolls a six-sided die that is biased so that rolling an even number has twice the probability of rolling an odd number.

the distribution of the discrete random variable X being the number showing of the die is:

X 1 2 3 4 5 6
p(X) 1/9 2/9 1/9 2/9 1/9 2/9

(i) Find the expected value and the variance of the distribution.

(ii) After how many rolls of the dice would you expect the sum of all values rolled to exceed a total of 100?

Answer.(i) E(X) = 3.7 and var (X) = 2.9.
(ii) Need 28 rolls to exceed 100.
 

17. Tracey and Alyssa are sisters who love to play netball. During the season they each keep a tally of the number of goals they score each game. At the end of the season, they share their match summaries which a shown below.

No of goals 0 1 2 3
Tracey 4 5 0 6
Alyssa 2 7 1 5

(i) Copy the table and calculate the probability of each player scoring 0, 1, 2 or 3 goals.

(ii) By calculating both the mean E(x) and the standard deviation for each player, conclude who is the better netballer and why.

 
  18. Jamie is playing a game of cards involving 10 cards - 6 of which are black and 4 are red. Some cards are heavier than others. The 10 cards are thrown in the air and Jamie counts the number of red cards which fall face-up. His results for the number of cards landing face-up and he determines the probabilities to be:
No. of red cards face-up 0 1 2 3 4
Probability 0.03 0.40 0.30 0.26 0.01

(i) What is the number of red cards Jamie should expect to fall face-up?

(ii) What is the variability in that estimate?

Answer.(i) Expect 1.82 red cards face up.
(ii) Var(X) = 0.89
  19. The number of butterflies in a garden at 10:00 am each morning is a random variable summarised in the following table:

No. of B/f 0 1 2 >2
Pr (B/f=x) 0.5 0.3 0.2 0

(i) How many butterflies could be expected in the garden at 10:00 am?

(ii) What is the standard deviation of the number of butterflies found at 10:00 am?

Answer.(i) Expect 0.7 butterflies.
(ii) SD = 0.78
 

20. The random variable X has the following probability distribution:

where k is a constant.

(i) Find the value of k.

(ii) Construct a table summarising the probability distribution of X.

(iii) Find E(X) and Var (X).

Answer.(i) k = 0.125.
(iii) E(X) = 3 and Var(X) = 1.
  21. Each morning outside my front door, the Crested Pigeons congregate waiting for seeds. They are usually in pairs and the following table summarises the probabilities I have estimated on the basis of many mornings of data collection.
 
No. of pairs of pigeons 4 5 6 7 8
Pr (pigeon) 0.1 0.1 0.4 0.2 0.2

(i) Find the number of pairs of Crested Pigeons I can expect to see tomorrow.

(ii) Find the variance in the number of pigeons who visit.

Answer.(i) Expected no. of pairs = 6.3.

(ii) Var(X) = 1.41.