1. The problem is to understand the hazard function in a Cox proportional hazards model, given by: $$\lambda(t|X,\beta) = \lambda_0(t) \exp(X' \beta)$$ 2. Here, $\lambda(t|X,\beta)$ is the hazard function at time $t$ given covariates $X$ and parameters $\beta$. 3. $\lambda_0(t)$ is the baseline hazard function, representing the hazard when all covariates are zero. 4. $X'$ is the transpose of the covariate vector $X$, and $\beta$ is the vector of regression coefficients. 5. The term $\exp(X' \beta)$ scales the baseline hazard according to the covariates. 6. This model assumes the hazard ratio between two individuals is proportional and constant over time. 7. The formula shows that the hazard at time $t$ is the baseline hazard multiplied by the exponential of the linear predictor $X' \beta$. Final answer: $$\lambda(t|X,\beta) = \lambda_0(t) \exp(X' \beta)$$