Statistics Solutions

[Steps Shown] Let Ti, i=1, ..., n be independent random variables. Denote hi(t) the corresponding hazard function of Ti and suppose that density function

Question: Let Ti, \(\mathrm{i}=1, \ldots, \mathrm{n}\) be independent random variables. Denote hi(t) the corresponding hazard function of Ti and suppose that density function fi(t) exists. Let \(\mathrm{T}=\min (\mathrm{T} 1, \mathrm{~T} 2\), ..., Tn). Denote \(\mathrm{h}_{\mathrm{T}}(\mathrm{t})\) the corresponding hazard function of \(\mathrm{T}\). Prove the following equality.

\[\mathrm{h}_{\mathrm{T}}(\mathrm{t})=\sum_{i=1}^{n} h_{i}(t)\]

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