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We derive analytical expressions for the probability density function (pdf) of the random mutual information between transmitted and received vector signals of a random space-time independent and identically distributed (i.i.d.) multiple-input multiple-output (MIMO) channel, assuming that the transmitted signals from the multiple antennas are Gaussian i.i.d.. We show that this pdf can be well approximated by a Gaussian distribution, and such a Gaussian approximation is based on expressions for the given pdfs mean and variance that we derive. We prove that at high signal-to-noise ratio (SNR), every factor of 2 increase in SNR leads to an increase in outage rate in the amount of min(M,N) bits, where M and N denote the number of transmit and receive antennas, respectively. A simple expression for the moment generating function (MGF) of the mutual information pdf is also provided, based on which we establish normality of the pdf, when both M and N are large, and the SNR is large.