Probability functions and cumulative distribution functionsEdexcel International A Level Maths: Subtopic test

1. 1
   The discrete random variable $X$ has probability function $P(X=x)=kx$ for $x=1,2,3,4$, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$10$
   • B$0.25$
   • C$0.2$
   • D$0.1$

   (b)

   Find $P(X\ge3)$.

   [1 mark]

   • A$0.3$
   • B$0.7$
   • C$0.4$
   • D$0.9$

   (c)

   Find $P(2\le X<4)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $X$ takes the values 1, 2, 3 and 4. Its cumulative distribution function is given by $F(1)=0.15$, $F(2)=0.40$, $F(3)=0.75$ and $F(4)=1$.

   (a)

   Find $P(X=3)$.

   [1 mark]

   • A$0.35$
   • B$0.75$
   • C$0.25$
   • D$0.15$

   (b)

   Find $P(X\ge2)$.

   [1 mark]

   • A$0.60$
   • B$0.15$
   • C$0.85$
   • D$0.25$

   (c)

   Find the probability that $X$ is either 2 or 4.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $X$ has probability function $P(X=x)=k(x^2+1)$ for $x=0,1,2,3$, where $k$ is a constant.

   (a)

   Find the value of $k$ and write down the probability function as a list of probabilities.

   [3 marks]

   (b)

   Find the cumulative distribution function $F(x)$ for each possible value of $x$, and hence find $P(X>1)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A fair four-sided die has faces numbered 1, 2, 3 and 4. The die is rolled twice and $X$ is the larger of the two scores (if the scores are equal, $X$ is that score).

   (a)

   (i) Show that $P(X=3)=\frac{5}{16}$.
   (ii) Find the probability function of $X$.
   (iii) Find the cumulative distribution function $F(x)$ for each possible value of $x$.

   [6 marks]

   (b)

   (i) Use $F(x)$ for two rolls to find the probability that the larger score is at least 3.
   (ii) The die is now rolled three times and $Y$ is the largest of the three scores. Show that $F(y)=\frac{y^3}{64}$ for $y=1,2,3,4$, and hence find $P(Y=3)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions