According to an expert, the inflation rate and the cash rate (the interest rate set by the Reserve Bank of Australia) in Australia in the next quarter is governed by the following joint probability density function: $$ \begin{matrix} \text{cash rate } \downarrow \diagdown \text{inflation rate } \longrightarrow & 1.00\% & 1.25\% & 1.50\% \ \hline 0.05\% & 0.06 & 0.10 & 0.00 \ 0.10\% & 0.04 & 0.50 & 0.10 \ 0.25\% & 0.00 & 0.10 & 0.10\end{matrix} $$ According to this expert, what is the expected value of the inflation rate conditional on cash rate staying at $0.10\%$ next quarter? Do all calculations to the highest precision and then round your final answer to two decimal points and enter it in percentage points but without the % sign.

Question

BlackTom AI · Accepted Answer

First, I will identify the joint probabilities given for each combination of cash rate and inflation rate. The table shows three possible cash rates (0.05%, 0.10%, 0.25%) across three inflation rates (1.00%, 1.25%, 1.50%), with the joint probabilities:
- Cash = 0.10% row: P(inflation=1.00%)=0.04, P(inflation=1.25%)=0.50, P(inflation=1.50%)=0.10.

Next, to find the expected inflation conditional on cash staying at 0.10%, I need the conditional distribution P(inflation | cash=0.10). This requires dividing each joint probability in the 0.10% row by P(cash=0.10).
Compute P(cash=0.10) by summing that row: 0.04 + 0.50 + 0.10 = 0.64.
Therefore, the conditional probabilities are:
- P(inflation=1.00% | cash=0.10) = 0.04 / 0.64 = 0.0625
- P(inflation=1.25% | cash=0.10) = 0.50 / 0.64 = 0.78125
- P(inflation=1.50% | cash=0.10) = 0.10 / 0.64 = 0.15625

Now, compute the conditional expectation of the inflation rate using these probabilities:
E[inflation | cash=0.10] = (0.0625)*(1.00) + (0.78125)*(1.25) + (0.15625)*(1.50).
Calculate each term: 0.0625, 0.78125×1.25 = 0.9765625, and 0.15625×1.50 = 0.234375.
Sum them to obtain 0.0625 + 0.9765625 + 0.234375 = 1.2734375.

Interpreting the units, this is in percentage points (since the inflation rates are given in percent). The instruction asks to round to two decimal places, so:
1.2734375% rounds to 1.27 percentage points.

For completeness, the precise value before rounding is 1.2734375, which confirms that the rounded result is 1.27.

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