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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.2

Name Date

Expected value of a discrete random variable Problem set A

Show the sum for E(X) each time. Give decimals exactly or to 3 significant figures. Check that each table adds to 1.

Difficulty: first skills applying them stretch

Skills

  1. The table shows the probability distribution of X, the number of passengers in a taxi.

    x12345P(X = x)0.320.280.200.120.08
    1. (a) Find E(X).
    2. (b) Can X take the value E(X)? Explain.

  2. A spinner pays a prize, Z pounds. The table shows the probability distribution of Z. Find E(Z).

    z102050100P(Z = z)0.500.300.150.05

  3. The random variable Y has distribution \(P(Y=y)=0.02(y^2-6y+18)\) for y = 0, 1, 2, 3.

    1. (a) Write the probability distribution of Y as a table.
    2. (b) Find E(Y).

  4. The table shows the probability distribution of Y.

    y1358P(Y = y)0.450.25k2k
    1. (a) Find the value of k.
    2. (b) Find P(Y ≤ 5).
    3. (c) Find E(Y).

  5. The table shows the probability distribution of Z, where k is a constant.

    z246810P(Z = z)0.12k0.283k0.2
    1. (a) Find the value of k.
    2. (b) Find P(Z ≤ 6).
    3. (c) Find P(Z > 4 | Z ≤ 8).

In context

  1. A bag holds 5 cherry sweets and 7 lime sweets. Two sweets are taken at random without replacement. X is the number of cherry sweets taken.

    1. (a) Find the probability distribution of X.
    2. (b) Find E(X).

  2. A bakery records X, the number of loaves a customer buys. The table shows its probability distribution.

    x0123P(X = x)0.150.450.300.10
    1. (a) Find E(X).
    2. (b) 200 customers are expected in a day. How many loaves should the shop expect to sell?

  3. The number of coffees, Y, bought by a customer has distribution \(P(Y=y)=0.01(y^2+3y+10)\) for y = 1, 2, 3, 4.

    1. (a) Write the probability distribution of Y as a table.
    2. (b) Find E(Y).

Stretch

  1. A drawer holds 6 black socks and 4 white socks. Three socks are taken at random without replacement. X is the number of white socks taken.

    1. (a) Find the probability distribution of X.
    2. (b) Find E(X).

  2. The table shows the probability distribution of Y, the number of rooms booked at a small hotel in an evening.

    y12345P(Y = y)k0.22k0.250.1
    1. (a) Find the value of k.
    2. (b) Find E(Y).
    3. (c) Find P(Y > E(Y)).

jmaths.xyz · AI SL · 13.2.2 Expected value of a discrete random variable