# Expected value of a random variable Max

You choose a uniformly random element, $a$, from $\{1, 2, . . . , 100\}$, and you choose a uniformly random element, $b$, from the same set $\{1, 2, . . . , 100\}$. ($a$ and $b$ are chosen independently)

Define the random variable X to be $X = max(a, b)$, where $X$ is the maximum of $a$ and $b$.

$\mathbb{E}(X) = \sum_{k=1}^{100} k \cdot \frac{1+2(k-1)}{100^2}$

I know that the formula for the expected value is $\mathbb{E}(X) = \sum_{}^{} k \cdot Pr(X = k)$.

I understand that the sample space is $100^2$, however I do not understand the numerator $1+2(k-1)$.

Thank you.

• HINT: What is the probability that $X=k$? Work through some examples (e.g. k=1, 2, 3, etc.) and see if you notice a pattern. Apr 17, 2018 at 20:57

When $X = k$, we have $3$ possibilities. We can have $a < b$ and $b = k$, we can have $a > b$ and $a = k$, and we can have $a = b = k$. There are $k - 1$ ways in which each of the first two scenarios can happen, since, in the first case for instance, we can have $a = 1, ..., k - 1$. This gives us $2(k-1) + 1$ distinct possibilities overall.

$X=k$ for all the cases $a = k$ and $b$ is chosen from $\{1,\dots, k-1\}$ so $k-1$ cases.

Same for $b=k$ so $k-1$ more cases.

Plus the case $a = k$ and $b=k$.

$2(k-1) +1$ favorable cases

Have a look at what $Pr(X=k)$ is, $X=k$ if:

• $a=b=k$ (which happens with probability $\frac{1}{100^2}$)
• $a=k$ (probability $\frac{1}{100}$), $b<k$ ($k-1$ possibilities so probability $\frac{k-1}{100}$), in the end that has a probability of $\frac{k-1}{100^2}$
• $b=k$ (probability $\frac{1}{100}$), $a<k$ ($k-1$ possibilities so probability $\frac{k-1}{100}$), in the end that has a probability of $\frac{k-1}{100^2}$

In the end, the total probability is the sum of the probabilities of those three cases.