Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 9

For a continuous random variable with probability density function $f_X$, its mean isprovided ${\int }_{-\infty }^{\infty }|x|f_X(x)dx<\infty$. Its variance is the expectation of the squared deviation from its mean:This is finite when $\mathbb{E}[X^2]<\infty$; otherwise it may be $+\infty$ if the finite mean exists.

Assume a finite second moment, so the following expectations are finite. Expand the square and use linearity of expectation:ThusThe same equality holds in the extended sense when the mean is finite and the second moment is infinite.

The normalization of a probability distribution requiresThe prescribed mean then givesAs $a,b>0$, these equations have the unique solution

The second moment isSubtracting the squared mean gives the variance

The random variable $Y=Z^2/2$ is nonnegative, so its cumulative distribution function is zero for $y<0$. If $y\ge 0$, thenThe last step uses the symmetry of the standard normal distribution, $\Phi (-z)=1-\Phi (z)$. Hence

Differentiate the cumulative distribution function for $y>0$ using the chain rule and the standard normal density:Thus a probability density function isThe value assigned at zero is immaterial. The probability density function is unbounded near zero, but $F_Y(0)=0$ and continuity of the cumulative distribution function there show that there is no point mass. This is a gamma distribution with shape $1/2$ and rate $1$.

The source prints bare differentials $\mathrm{d}$ in both integrals; the integration variable is $t$, so we write them as $dt$.^[ref]

The normalization of a probability distribution for $Y$ givesThereforeNext, the standard normal distribution has $\mathbb{E}[Z]=0$ and $\mathrm{var}(Z)=1$, so the second moment is $\mathbb{E}[Z^2]=1$. Since $Y=Z^2/2$, linearity of expectation yields $\mathbb{E}[Y]=1/2$. Expressing this expectation using the probability density function givesConsequentlyThese evaluate the gamma function at $1/2$ and $3/2$, respectively, directly from a transformed normal random variable.