Question

Below is an incomplete probability distribution for a random variable X.

x 1 2 3 4 5 6

f(x) 0.15 0.25 0.33 0.09

(a) If f(4) = P(X = 4) = 2P(X = 5), complete the probability distribution above.

(b)Construct a probability histogram of this distribution.

(c) Find the cumulative distribution of the random variable X.

(d)Construct a graph of the cumulative distribution.

Answer 1

(a)

$$0.15 +0.25+ 0.33+ 0.09+k=1$$

$$k=1-0.82=0.18$$

$$P(X=5)=\dfrac{1}{2}P(X=4)=\dfrac{1}{2}(0.09)=0.045$$

$$P(X=6)=k-P(X=5)=0.18-0.045=0.135$$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & 0.15 & 0.25 & 0.33 & 0.09 & 0.045 & 0.135 \end{array}$$

(b)

(c)

$$F(x) = \begin{cases} 0 &x<1 \\ 0.15 &1\le x<2\\ 0.40 &2\le x<3\\ 0.73 & 3\le x<4\\ 0.82 & 4\le x<5\\ 0.865 & 5\le x<6\\ 1 & x\ge6\\ \end{cases}$$

(d)