import torch


def marginal_prob_std(t, sigma, device):
    """Compute the mean and standard deviation of $p_{0t}(x(t) | x(0))$."""
    t = torch.tensor(t, device=device)
    beta_1 = 20.0
    beta_0 = 0.1
    log_mean_coeff = -0.25 * t**2 * (beta_1 - beta_0) - 0.5 * t * beta_0
    alpha_t = torch.exp(log_mean_coeff)
    std = torch.sqrt(1.0 - torch.exp(2.0 * log_mean_coeff))
    return alpha_t, std
