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We propose a new unequal error protection (UEP) scheme for Luby Transform (LT) codes over additive white Gaussian noise (AWGN) channels. In contrast to the state-of-the-art UEP scheme for LT codes where the effective code rates are exploited for UEP, we characterize the variable-node degree distribution to achieve UEP for LT codes. For performance analysis, we compare the lower bounds on bit error rate for our proposed scheme and a conventional UEP-based LT codes over AWGN channels for various parameters. Moreover, it is demonstrated through numerical examples that our proposed UEP scheme has better performance than the conventional UEP-based LT codes over a wide range of code rates and channel conditions.