Question

Human height is well described by a normal random variable. Suppose a simple random sample of two individuals is taken, and $X_1$ and $X_2$ are their heights. The random variables $X_1$ and $X_2$ are i.i.d. normal random variables with common probability density function given by $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$$ Let $F(x)$ be their common CDF. (a) (2 pts) Write an expression for the probability that $X_1 \leq x_1$ and $X_2 \leq x_2$, in terms of the function $F(x)$. (b) (2 pts) The expression in the previous part is called the joint CDF of $X_1$ and $X_2$. If this function is denoted by $G(x_1,x_2)$ then it can be proved that the joint probability density function of $X_1$ and $X_2$, denoted by $g(x_1,x_2)$, is given by $$g(x_1, x_2) = \frac{\partial^2 G}{\partial x_1 \partial x_2}$$ Calculate the joint probability density function. (c) (2 pts) Suppose the heights of the two individuals are 160 cm and 170 cm. Find the value of the joint PDF for this sample, as a function of the population mean $\mu$. (d) (2 pts) The function in part (c) is referred to as a likelihood function. It gives the value of the joint PDF when evaluated at the sampled data points, for different values of the population mean $\mu$. The maximum likelihood estimate for the population mean height is the value of $\mu$ that maximizes this likelihood function. In other words, it is the value of that makes the joint PDF at the sampled data as large as possible. Find this value of $\mu$.

          Human height is well described by a normal random variable. Suppose a simple random sample of two
individuals is taken, and $X_1$ and $X_2$ are their heights. The random variables $X_1$ and $X_2$ are i.i.d. normal
random variables with common probability density function given by
$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)}$$
Let $F(x)$ be their common CDF.

(a) (2 pts) Write an expression for the probability that $X_1 \leq x_1$ and $X_2 \leq x_2$, in terms of the function
$F(x)$.

(b) (2 pts) The expression in the previous part is called the joint CDF of $X_1$ and $X_2$. If this function is
denoted by $G(x_1,x_2)$ then it can be proved that the joint probability density function of $X_1$ and $X_2$,
denoted by $g(x_1,x_2)$, is given by
$$g(x_1, x_2) = \frac{\partial^2 G}{\partial x_1 \partial x_2}$$
Calculate the joint probability density function.

(c) (2 pts) Suppose the heights of the two individuals are 160 cm and 170 cm. Find the value of the joint
PDF for this sample, as a function of the population mean $\mu$.

(d) (2 pts) The function in part (c) is referred to as a likelihood function. It gives the value of the joint
PDF when evaluated at the sampled data points, for different values of the population mean $\mu$. The
maximum likelihood estimate for the population mean height is the value of $\mu$ that maximizes this
likelihood function. In other words, it is the value of that makes the joint PDF at the sampled data
as large as possible. Find this value of $\mu$.

human height is well described by a normal random variable suppose a simple random sample of two individuals is taken and x1 and x2 are their heights the random variables x1 and x2 are i 85084

Added by Clinton M.

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