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Consider a density p (x1, x2) zero everywhere, except on the set { (x1, x2) R2: x2 = x12 }, and is unimodal on this line with a maximum at (1, 1). The marginal density on x1 is unimodal with a maximum at x1 =1, but the marginal density on x2 can have two peaks at x2 =1 and x2 =0. Depending on how the density lies on the line, the peak at x2 =0 can be larger than the peak at x2 =1; hence, a marginal density can be misleading. Consequently, it is generally better to maximize a joint density to avoid this kind of situation.
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Consider a density p (x1, x2) zero everywhere, except on the set { (x1, x2) R2: x2 = x12 }, and is unimodal on this line with a maximum at (1, 1). The marginal density on x1 is unimodal with a maximum at x1 =1, but the marginal density on x2 can have two peaks at x2 =1 and x2 =0. Depending on how the density lies on the line, the peak at x2 =0 can be larger than the peak at x2 =1; hence, a marginal density can be misleading. Consequently, it is generally better to maximize a joint density to avoid this kind of situation.
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