Expectation and variance of discrete random variablesEdexcel International A Level Maths: Subtopic test

1. 1
   The discrete random variable $X$ has probability distribution $P(X=0)=0.2$, $P(X=1)=0.3$, $P(X=2)=0.4$ and $P(X=3)=0.1$.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$1.5$
   • B$2.8$
   • C$1.4$
   • D$0.25$

   (b)

   Find $\mathrm{Var}(X)$.

   [1 mark]

   • A$2.8$
   • B$4.76$
   • C$1.4$
   • D$0.84$

   (c)

   Find $E(3X+2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variable $W$ has $E(W)=10$ and $\mathrm{Var}(W)=6$. The random variable $V$ is defined by $V=3W-4$.

   (a)

   Find $E(V)$.

   [1 mark]

   • A$26$
   • B$34$
   • C$30$
   • D$18$

   (b)

   Find $\mathrm{Var}(V)$.

   [1 mark]

   • A$18$
   • B$54$
   • C$14$
   • D$50$

   (c)

   Find $E(W^2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ takes the values 1, 2, 3 and 4 with $P(X=1)=0.1$, $P(X=2)=a$, $P(X=3)=0.4$ and $P(X=4)=b$, where $a$ and $b$ are constants. It is given that $E(X)=2.9$.

   (a)

   Find the values of $a$ and $b$.

   [3 marks]

   (b)

   Find $\mathrm{Var}(X)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a game, a player rolls a fair six-sided die once and wins a prize, $X$ pounds, that depends on the score. A score of 6 wins 8 pounds, a score of 4 or 5 wins 3 pounds, and a score of 1, 2 or 3 wins nothing.

   (a)

   (i) Write down the probability distribution of $X$.
   (ii) Find $E(X)$.
   (iii) Find $\mathrm{Var}(X)$.

   [6 marks]

   (b)

   The player pays 2 pounds to play, so the net gain is $Y=X-2$ pounds.
   (i) Find $E(Y)$ and $\mathrm{Var}(Y)$.
   (ii) The organiser changes the entry fee to 4 pounds. Find the new expected net gain per game, and the entry fee that would make the game fair (expected net gain of zero).
   (iii) Comment on whether the game is favourable to the player at the original fee of 2 pounds.

   [6 marks]

   Total for question 4: 12 marks

End of questions