Expectation of the binomial distribution, where the number of trials is a random variable

Question

Suppose

• $Y_1 \sim Binomial(X_1,\delta)$
• $Y_2 \sim Binomial(X_2,\delta)$
• $X_1>0$ and $X_2>0$ are random variables
• $E[X_1]>E[X_2]$

Then does the following hold? $E[Y_1]>E[Y_2]$

Answer 1

By the law of iterated expectation, we have $$E[Y_i]=E[E[Y_i|X_i]]=\delta E[X_i]$$

Since $E[X_1] > E[X_2]$, if $\delta >0$, we have

$$\delta E[X_1] > \delta E[X_2]$$

Hence $E[Y_1]> E[Y_2]$.