Question

let the discrete random variable, x, have the following probability mass function

f(x)= c|x-2|, x=-2,-1,0,1,2,3

a. find the constant such that fx is a valid pmf

b. find P(|X-1|>1)

c. find the expected value of X

d. find the expected value of X^2

Answer #1

X | P(X) | |

-2 | 4c | |

-1 | 3c | |

0 | 0 | |

1 | 1c | |

2 | 0 | |

3 | c | |

total | 9c |

a)

total sum 9c must be 1 to be a valid PDF

so, 9c=1

c=1/9

b)

P(|X-1|>1)

|X-1| | P(X) |

3 | =4/9 |

2 | =3/9 |

1 | =0 |

0 | =1/9 |

1 | 0 |

2 | =1/9 |

so, P(|X-1|>1 = P(|X-1|=2)+P(|X-1|=3) = 3/9+4/9=7/9

c)

X | P(X) | X*P(X) |

-2 | 4/9 | -0.88889 |

-1 | 1/3 | -0.33333 |

0 | 0 | 0 |

1 | 1/9 | 0.111111 |

2 | 0 | 0 |

3 | 1/9 | 0.333333 |

P(X) | X*P(X) | |

total sum = | 1 | -0.77778 |

mean = E[X] = Σx*P(X) = |
-0.77778 |
||

d)

X² | P(X) | X²*P(X) |

4 | 4/9 | 1.777778 |

1 | 1/3 | 0.333333 |

0 | 0 | 0 |

1 | 1/9 | 0.1111 |

4 | 0 | 0 |

9 | 1/9 | 1 |

P(X) | X²*P(X) | |

total sum = | 1 | 3.2222 |

E[X²] = Σx²*P(X) = 3.2222

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