Read a marginal distribution
A component of a jointly normal vector is normal. Its mean comes from the corresponding mean-vector entry, and its variance from the diagonal of the covariance matrix.
Probability & statistics / English
Multivariate Normal Distribution | Probabilities
math et al · YouTube · 8:24
YouTube · math et alOriginal video
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Reduce multivariate normal probability questions to univariate normal distributions using means and covariances.
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0:00Read a marginal distribution3:00A linear combination stays normal6:00Keep the covariance termsLearning script
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A component of a jointly normal vector is normal. Its mean comes from the corresponding mean-vector entry, and its variance from the diagonal of the covariance matrix.
For the given jointly normal X₂ and X₃, the combination 5X₂+4X₃ has mean and variance , hence standard deviation 18.
For a row coefficient vector a, the variance of aX is aΣaᵀ. Include correlations before standardizing with the resulting standard deviation; marginal normality alone does not guarantee every combination is normal.
Knowledge cards
A linear combination stays normal
For the given jointly normal X₂ and X₃, the combination 5X₂+4X₃ has mean and variance , hence standard deviation 18.
Keep the covariance terms
For a row coefficient vector a, the variance of aX is aΣaᵀ. Include correlations before standardizing with the resulting standard deviation; marginal normality alone does not guarantee every combination is normal.
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A component of a jointly normal vector is normal. Its mean comes from the corresponding mean-vector entry, and its variance from the diagonal of the covariance matrix. For the given jointly normal X₂ and X₃, the combination 5X₂+4X₃ has mean and variance , hence standard deviation 18. For a row coefficient vector a, the variance of aX is aΣaᵀ. Include correlations before standardizing with the resulting standard deviation; marginal normality alone does not guarantee every combination is normal.
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