Expected Value — Practice Problems and Worked Solutions

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17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the expected value of X.

   $$P(X=0)=\frac{1}{4},P(X=1)=\frac{1}{2},P(X=2)=\frac{1}{4}$$

   Answer: $1$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=1$$
2. 2.

   Find the expected value of X.

   $$P(X=1)=\frac{1}{3},P(X=2)=\frac{1}{3},P(X=3)=\frac{1}{3}$$

   Answer: $2$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=2$$
3. 3.

   Find the expected value of X.

   $$P(X=0)=\frac{1}{2},P(X=2)=\frac{1}{4},P(X=4)=\frac{1}{4}$$

   Answer: $\frac{3}{2}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{3}{2}$$
4. 4.

   Find the expected value of X.

   $$P(X=-1)=\frac{2}{5},P(X=1)=\frac{3}{5}$$

   Answer: $\frac{1}{5}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{1}{5}$$
5. 5.

   Find the expected value of X.

   $$P(X=1)=\frac{1}{2},P(X=5)=\frac{2}{5},P(X=10)=\frac{1}{10}$$

   Answer: $\frac{7}{2}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{7}{2}$$
6. 6.

   Find the expected value of X.

   $$P(X=0)=\frac{1}{6},P(X=3)=\frac{1}{2},P(X=6)=\frac{1}{3}$$

   Answer: $\frac{7}{2}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{7}{2}$$
7. 7.

   Find the expected value of X.

   $$P(X=0)=\frac{1}{3},P(X=1)=\frac{2}{3}$$

   Answer: $\frac{2}{3}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{2}{3}$$
8. 8.

   Find the expected value of X.

   $$P(X=2)=\frac{1}{4},P(X=4)=\frac{1}{4},P(X=6)=\frac{1}{2}$$

   Answer: $\frac{9}{2}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{9}{2}$$
9. 9.

   Find the expected value of X.

   $$P(X=1)=\frac{3}{5},P(X=2)=\frac{2}{5}$$

   Answer: $\frac{7}{5}$

   Step-by-step solution

   1. $$E[X]=\sum _{x}xP(X=x)$$
   2. $$=\frac{7}{5}$$
10. 10.

    Find the expected value of X.

    $$P(X=0)=\frac{1}{5},P(X=5)=\frac{2}{5},P(X=10)=\frac{2}{5}$$

    Answer: $6$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=6$$
11. 11.

    Find the expected value of X.

    $$P(X=-2)=\frac{1}{4},P(X=0)=\frac{1}{2},P(X=2)=\frac{1}{4}$$

    Answer: $0$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=0$$
12. 12.

    Find the expected value of X.

    $$P(X=1)=\frac{1}{3},P(X=3)=\frac{1}{3},P(X=9)=\frac{1}{3}$$

    Answer: $\frac{13}{3}$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=\frac{13}{3}$$
13. 13.

    Find the expected value of X.

    $$P(X=0)=\frac{3}{4},P(X=4)=\frac{1}{4}$$

    Answer: $1$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=1$$
14. 14.

    Find the expected value of X.

    $$P(X=5)=\frac{1}{2},P(X=10)=\frac{3}{10},P(X=20)=\frac{1}{5}$$

    Answer: $\frac{19}{2}$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=\frac{19}{2}$$
15. 15.

    Find the expected value of X.

    $$P(X=-1)=\frac{1}{6},P(X=0)=\frac{1}{3},P(X=3)=\frac{1}{2}$$

    Answer: $\frac{4}{3}$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=\frac{4}{3}$$
16. 16.

    Find the expected value of X.

    $$P(X=2)=\frac{2}{5},P(X=8)=\frac{3}{5}$$

    Answer: $\frac{28}{5}$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=\frac{28}{5}$$
17. 17.

    Find the expected value of X.

    $$P(X=0)=\frac{1}{8},P(X=1)=\frac{1}{2},P(X=4)=\frac{3}{8}$$

    Answer: $2$

    Step-by-step solution

    1. $$E[X]=\sum _{x}xP(X=x)$$
    2. $$=2$$

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