The tip $ X $ (in Euros) that a waiter in a cafe / restaurant receives from a guest is distributed as follows:
If the guest orders a hot meal, $X$ has expectation $2$ and variance $2$
If the guest does not order a hot meal (for example, just a drink), $X$ has expectation $0, 5$ and variance $0, 3$
The probability $p$ that a guest orders a hot meal is $ 0, 3$.
a) Calculate the expected value $\mu_X$ and the variance $\sigma^2$ of $X$.
b) Determine the (approximative) propability that the waiter of $n=100$ guesrs gets less that $80$ Euro tips.
At the question a) do we have to define a new random variable that is a linear combination of the random variables $X_1, X_2$, where these two describe the tips if the guest orders a meal or a drink respectively?