Discrete random variables Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Discrete random variables questions. See exactly how to solve problems on discrete random variable, probability distribution, probabilities sum to 1, cumulative probability.

discrete random variableprobability distributionprobabilities sum to 1cumulative probabilitydiscrete uniform distributionprobability
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
A discrete random variable XX has the probability distribution given by P(X=0)=0.1P(X=0)=0.1, P(X=1)=0.3P(X=1)=0.3, P(X=2)=kP(X=2)=k, P(X=3)=0.2P(X=3)=0.2. Find the value of kk.

Worked solution

  1. Use that all the probabilities sum to 1

    ∑P(X=x)=1\sum P(X=x) = 1

    For a discrete random variable the probabilities of every outcome add to 1.

  2. Substitute the entries of the distribution

    0.1+0.3+k+0.2=10.1 + 0.3 + k + 0.2 = 1

    Write the total of all the listed probabilities.

  3. Solve the equation for k

    k=0.4k = 0.4

    Rearrange to find the unknown constant.

Answer
k=0.4k = 0.4
Question 2
2 markseasy
The random variable XX has P(X=1)=0.15P(X=1)=0.15, P(X=2)=kP(X=2)=k, P(X=3)=0.25P(X=3)=0.25 and P(X=4)=0.3P(X=4)=0.3. Find the value of kk.

Worked solution

  1. Use that all the probabilities sum to 1

    ∑P(X=x)=1\sum P(X=x) = 1

    For a discrete random variable the probabilities of every outcome add to 1.

  2. Substitute the entries of the distribution

    0.15+k+0.25+0.3=10.15 + k + 0.25 + 0.3 = 1

    Write the total of all the listed probabilities.

  3. Solve the equation for k

    k=0.3k = 0.3

    Rearrange to find the unknown constant.

Answer
k=0.3k = 0.3
Question 3
2 markseasy
A discrete random variable XX takes the values 0,1,20,1,2 with P(X=0)=kP(X=0)=k, P(X=1)=0.5P(X=1)=0.5 and P(X=2)=0.2P(X=2)=0.2. Find the value of kk.

Worked solution

  1. Use that all the probabilities sum to 1

    ∑P(X=x)=1\sum P(X=x) = 1

    For a discrete random variable the probabilities of every outcome add to 1.

  2. Substitute the entries of the distribution

    k+0.5+0.2=1k + 0.5 + 0.2 = 1

    Write the total of all the listed probabilities.

  3. Solve the equation for k

    k=0.3k = 0.3

    Rearrange to find the unknown constant.

Answer
k=0.3k = 0.3
Question 4
2 markseasy
The random variable XX has P(X=2)=0.2P(X=2)=0.2, P(X=3)=0.2P(X=3)=0.2, P(X=4)=kP(X=4)=k and P(X=5)=0.1P(X=5)=0.1. Find the value of kk.

Worked solution

  1. Use that all the probabilities sum to 1

    ∑P(X=x)=1\sum P(X=x) = 1

    For a discrete random variable the probabilities of every outcome add to 1.

  2. Substitute the entries of the distribution

    0.2+0.2+k+0.1=10.2 + 0.2 + k + 0.1 = 1

    Write the total of all the listed probabilities.

  3. Solve the equation for k

    k=0.5k = 0.5

    Rearrange to find the unknown constant.

Answer
k=0.5k = 0.5
Question 5
2 markseasy
A discrete random variable XX has P(X=1)=0.35P(X=1)=0.35, P(X=2)=0.25P(X=2)=0.25 and P(X=3)=kP(X=3)=k. Find the value of kk.

Worked solution

  1. Use that all the probabilities sum to 1

    ∑P(X=x)=1\sum P(X=x) = 1

    For a discrete random variable the probabilities of every outcome add to 1.

  2. Substitute the entries of the distribution

    0.35+0.25+k=10.35 + 0.25 + k = 1

    Write the total of all the listed probabilities.

  3. Solve the equation for k

    k=0.4k = 0.4

    Rearrange to find the unknown constant.

Answer
k=0.4k = 0.4

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