- marginal probability mass function of X and Y ,
- P(x≤1, Y=2),
- P(X≤1)

X/Y | 1 | 2 | 3 | 4 | 5 | 6 |

0 | 0 | 0 | 1/32 | 2/32 | 2/32 | 3/32 |

1 | 1/16 | 1/16 | 1/8 | 1/8 | 1/8 | 1/8 |

2 | 1/32 | 1/32 | 1/64 | 1/64 | 1/64 | 1/64 |

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Login NowA variable whose value id determined by the outcome of a random experiment is called a random variable.

Given,

xy | 1 | 2 | 3 | 4 | 5 | 6 | P(X=x) |

0 | 0 | 0 | 1/32 | 2/32 | 2/32 | 3/32 | 8/32 |

1 | 1/16 | 1/16 | 1/8 | 1/8 | 1/8 | 1/8 | 5/8 |

2 | 1/32 | 1/32 | 1/64 | 1/64 | 1/64 | 1/64 | 1/8 |

P(Y=y) | 3/32 | 3/32 | 1/64 | 13/64 | 13/64 | 13/64 | sum = 1 |

**Solution i):**

Marginal pmf of x = p(X = xi) = \(\sum_{j=1}^n p(x_i y_i)\)

∴P(X = 0) = P(X=0, Y=1) + P(X = 0, Y = 2) + P(X = 0, Y = 3) + P(X = 0, Y = 4) + P(X = 0, Y = 5) + P(X = 0, Y = 6)

= 8/32

∴P(X = 1) = P(X=1, Y=1) + P(X = 1, Y = 2) + P(X = 1, Y = 3) + P(X = 1, Y = 4) + P(X = 1, Y = 5) + P(X = 1, Y = 6)

= 5/8

∴P(X = 2) = P(X=2, Y=1) + P(X = 2, Y = 2) + P(X = 2, Y = 3) + P(X = 2, Y = 4) + P(X = 2, Y = 5) + P(X = 2, Y = 6)

= 1/8

Also, Marginal pmf of Y = P(Y=yi) = \(\sum_{i=1}^n p(x_i y_i)\)

∴P(Y = 1) = P(X=0, Y=1) + P(X = 1, Y = 1) + P(X = 1, Y = 2)

= 3/32

Similarly,

P(Y = 2) = 3/32

P(Y = 3) = 11/64

P(Y = 4) = 13/64

P(Y = 5) = 13/64

P(Y = 6) = 15/64

**Solution ii):**

P(X ≤ 1, Y = 2)

= P(X = 0, Y= 2) + P(X = 1, Y = 2)

= 0 + 1/16

= 1/16

**Solution ii)**

P(X ≤ 1)

= P(X = 0) + P(X = 1)

= 8/32 + 5/8

= 7/8

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## Discussion